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first appeared in 


a series of new books describing, in 
language suitable for the general non- 
specialist reader, the present position 
in many branches of modern science. 
The series is edited by Dr C. P. Snow, 
and is published by 












BEFORE you begin reading, you rightly expect some 
simple questions to be answered. For what purpose has 
this book been written? Who is the imaginary reader 
for whom it is meant? 

It is difficult to begin by answering these questions 
clearly and convincingly. This would be much easier, 
though quite superfluous, at the end of the book. We 
find it simpler to say just what this book does not 
intend to be. We have not written a textbook of 
physics. Here is no systematic course in elementary 
physical facts and theories. Our intention was rather 
to sketch in broad outline the attempts of the human 
mind to find a connection between the world of ideas 
and the world of phenomena. We have tried to show 
the active forces which compel science to invent ideas 
corresponding to the reality of our world. But our 
representation had to be simple. Through the maze of 
facts and concepts we had to choose some highway 
which seemed to us most characteristic and significant. 
Facts and theories not reached by this road had to be 
omitted. We were forced, by our general aim, to make 
a definite choice of facts and ideas. The importance of a 
probletn should not be judged by the number of pages 
devoted to it. Some essential lines of thought have been 
left out, not because they seemed to us unimportant, 
but because they do not lie along the road we have 


Whilst writing the book we had long discussions as to 
the characteristics of our idealized reader and worried 
a good deal about him. We had him making up for a 
complete lack of any concrete knowledge of physics 
and mathematics by quite a great number of virtues. 
We found him interested in physical and philosophical 
ideas and we were forced to admire the patience with 
which he struggled through the less interesting and 
more difficult passages. He realized that in order to 
understand any page he must have read the preceding 
ones carefully. He knew that a scientific book, even 
though popular, must not be read in the same way as 
a novel. 

The book is a simple chat between you and us. You 
may find it boring or interesting, dull or exciting, but 
our aim will be accomplished if these pages give you 
some idea of the eternal struggle of the inventive 
human mind for a fuller understanding of the laws 
governing physical phenomena. 

A. E. 

L. I. 


WE WISH to thank all those who have so kindly helped 
us with the preparation of this book, in particular : 

Professors A. G. Shenstone, Princeton, N.J., and St 
Loria, Lwow, Poland, for photographs on plate III. 

I. N. Steinberg for his drawings. 

Dr M. Phillips for reading the manuscript and for 
her very kind help. 

A. E. 

L. I. 



The great mystery story page 3 

The first clue 5 

Vectors 12 

The riddle of motion 19 

One clue remains 34 

Is heat a substance? 38 

The switchback 47 

The rate of exchange 51 

The philosophical background 55 

The kinetic theory of matter 59 


The two electric fluids 71 

The magnetic fluids 83 

The first serious difficulty 87 

The velocity of light 94 

Light as substance 97 

The riddle of colour 100 

What is a wave? 104 

The wave theory of light no 

Longitudinal or transverse light waves? 120 

Ether and the mechanical view 123 


The field as representation 1 29 

The two pillars of the field theory 142 


The reality of the field page 148 

Field and ether 156 

The mechanical scaffold 160 

Ether and motion 172 

Time, distance, relativity 186 

Relativity and mechanics 202 

The time-space continuum 209 

General relativity 220 

Outside and inside the lift 226 

Geometry and experiment 235 

General relativity and its verification 249 

Field and matter 255 


Continuity, discontinuity 263 

Elementary quanta of matter and electricity 265 

The quanta of light 272 

Light spectra 280 

The waves of matter 286 

Probability waves 294 

Physics and reality 310 

Index 3 i 7 


Plate i. Brownian movement facing P a S e 66 

n. Diffraction of light 118 

in. Spectral lines, diffraction of X-rays and of 

electronic waves 286 



The great mystery story The fast clue Vectors The riddle 
of motion One clue remains Is heat a substance? The 
switchback The rate of exchange The philosophical back- 
ground The kinetic theory of matter 


IN IMAGINATION there exists the perfect mystery story. 
Such a story presents all the essential clues, and com- 
pels us to form our own theory of the case. If we 
follow the plot carefully, we arrive at the complete 
solution for ourselves just before the author's disclosure 
at the end of the book. The solution itself, contrary to 
those of inferior mysteries, does not disappoint us; more- 
over, it appears at the very moment we expect it. 

Can we liken the reader of such a book to the scientists, 
who throughout successive generations continue to seek 
solutions of the mysteries in the book of nature? The 
comparison is false and will have to be abandoned later, 
but it has a modicum of justification which may be 
extended and modified to make it more appropriate to 
the endeavour of science to solve the mystery of the 

This great mystery story is still unsolved. We cannot 


even be sure that it has a final solution. The reading 
has already given us much; it has taught us the rudi- 
ments of the language of nature; it has enabled us to 
understand many of the clues, and has been a source 
of joy and excitement in the oftentimes painful ad- 
vance of science. But we realize that in spite of all the 
volumes read and understood we are still far from a 
complete solution, if, indeed, such a thing exists at all. 
At every stage we try to find an explanation consistent 
with the clues already discovered. Tentatively accepted 
theories have explained many of the facts, but no 
general solution compatible with all known clues has 
yet been evolved. Very often a seemingly perfect theory 
has proved inadequate in the light of further reading. 
New facts appear, contradicting the theory or unex- 
plained by it. The more we read, the more fully do we 
appreciate the perfect construction of the book, even 
though a complete solution seems to recede as we ad- 

In nearly every detective novel since the admirable 
stories of Conan Doyle there comes a time when the 
investigator has collected all the facts he needs for at 
least some phase of his problem. These facts often seem 
quite strange, incoherent, and wholly unrelated. The 
great detective, however, realizes that no further in- 
vestigation is needed at the moment, and that only pure 
thinking will lead to a correlation of the facts collected. 
So he plays his violin, or lounges in his armchair en- 
joying a pipe, when suddenly, by Jove, he has it ! Not 
only does he have an explanation for the clues at hand, 


but he knows that certain other events must have 
happened. Since he now knows exactly where to look 
for it, he may go out, if he likes, to collect further con- 
firmation for his theory. 

The scientist reading the book of nature, if we may 
be allowed to repeat the trite phrase, must find the 
solution for himself; for he cannot, as impatient readers 
of other stories often do, turn to the end of the book. 
In our case the reader is also the investigator, seeking 
to explain, at least in part, the relation of events to 
their rich context. To obtain even a partial solution the 
scientist must collect the unordered facts available and 
make them coherent and understandable by creative 

It is our aim, in the following pages, to describe in 
broad outline that work of physicists which corre- 
sponds to the pure thinking of the investigator. We 
shall be chiefly concerned with the role of thoughts 
and ideas in the adventurous search for knowledge of 
the physical world. 


Attempts to read the great mystery story are as old as 
human thought itself. Only a little over three hundred 
years ago, however, did scientists begin to understand 
the language of the story. Since that time, the age of 
Galileo and Newton, the reading has proceeded rapidly. 
Techniques of investigation, systematic methods of find- 
ing and following clues, have been developed. Some of 


the riddles of nature have been solved, although many 
of the solutions have proved temporary and superficial 
in the light of further research. 

A most fundamental problem, for thousands of years 
wholly obscured by its complications, is that of motion. 
All those motions we observe in nature that of a 
stone thrown into the air, a ship sailing the sea, a cart 
pushed along the street are in reality very intricate. 
To understand these phenomena it is wise to begin 
with the simplest possible cases, and proceed gradually 
to the more complicated ones. Consider a body at rest, 
where there is no motion at all. To change the position 
of such a body it is necessary to exert some influence 
upon it, to push it or lift it, or let other bodies, such as 
horses or steam engines, act upon it. Our intuitive idea 
is that motion is connected with the acts of pushing, 
lifting or pulling. Repeated experience would make us 
risk the further statement that we must push harder if 
we wish to move the body faster. It seems natural to 
conclude that the stronger the action exerted on a body, 
the greater will be its speed. A four-horse carriage 
goes faster than a carriage drawn by only two horses. 
Intuition thus tells us that speed is essentially connected 
with action. 

It is a familiar fact to readers of detective fiction 
that a false clue muddles the story and postpones the 
solution. The method of reasoning dictated by intui- 
tion was wrong and led to false ideas of motion which 
were held for centuries. Aristotle's great authority 
throughout Europe was perhaps the chief reason for 


the long belief in this intuitive idea. We read in the 
Mechanics , for two thousand years attributed to him: 

The moving body comes to a standstill when the force 
which pushes it along can no longer so act as to push it. 

The discovery and use of scientific reasoning by 
Galileo was one of the most important achievements 
in the history of human thought, and marks the real 
beginning of physics. This discovery taught us that 
intuitive conclusions based on immediate observation 
are not always to be trusted, for they sometimes lead 
to the wrong clues. 

But where does intuition go wrong? Can it possibly 
be wrong to say that a carriage drawn by four horses 
must travel faster than one drawn by only two? 

Let us examine the fundamental facts of motion 
more closely, starting with simple everyday experiences 
familiar to mankind since the beginning of civilization 
and gained in the hard struggle for existence. 

Suppose that someone going along a level road with 
a pushcart suddenly stops pushing. The cart will go 
on moving for a short distance before coming to rest. 
We ask: how is it possible to increase this distance? 
There are various ways, such as oiling the wheels, and 
making the road very smooth. The more easily the 
wheels turn, and the smoother the road, the longer the 
cart will go on moving. And just what has been done 
by the oiling and smoothing? Only this: the external 
influences have been made smaller. The effect of what 
is called friction has been diminished, both in the 
wheels and between the wheels and the road. This is 


already a theoretical interpretation of the observable 
evidence, an interpretation which is, in fact, arbitrary. 
One significant step farther and we shall have the right 
clue. Imagine a road perfectly smooth, and wheels 
with no friction at all. Then there would be nothing 
to stop the cart, so that it would run for ever. This 
conclusion is reached only by thinking of an idealized 
experiment, which can never be actually performed, 
since it is impossible to eliminate all external influences. 
The idealized experiment shows the clue which really 
formed the foundation of the mechanics of motion. 

Comparing the two methods of approaching the 
problem, we can say: the intuitive idea is the greater 
the action, the greater the velocity. Thus the velocity 
shows whether or not external forces are acting on 
a body. The new clue found by Galileo is : if a body 
is neither pushed, pulled, nor acted on in any other 
way, or, more briefly, if no external forces act on a 
body, it moves uniformly, that is, always with the 
same velocity along a straight line. Thus, the velocity 
does not show whether or not external forces are act- 
ing on a body. Galileo's conclusion, the correct one, 
was formulated a generation later by Newton as the 
law of inertia. It is usually the first thing about physics 
which we learn by heart in school, and some of us may 
remember it : 

Every body perseveres in its state of rest, or of uniform 
motion in a right line, unless it is compelled to change that 
state by forces impressed thereon. 

We have seen that this law of inertia cannot be 


derived directly from experiment, but only by specula- 
tive thinking consistent with observation. The idealized 
experiment can never be actually performed, although 
it leads to a profound understanding of real experi- 

From the variety of complex motions in the world 
around us we choose as our first example uniform mo- 
tion. This is the simplest, because there are no external 
forces acting. Uniform motion can, however, never be 
realized; a stone thrown from a tower, a cart pushed 
along a road can never move absolutely uniformly be- 
cause we cannot eliminate the influence of external 

In a good mystery story the most obvious clues 
often lead to the wrong suspects. In our attempts to 
understand the laws of nature we find, similarly, that 
the most obvious intuitive explanation is often the 
wrong one. 

Human thought creates an ever-changing picture of 
the universe. Galileo's contribution was to destroy the 
intuitive view and replace it by a new one. This is the 
significance of Galileo's discovery. 

But a further question concerning motion arises im- 
mediately. If the velocity is no indication of the ex- 
ternal forces acting on a body, what is? The answer to 
this fundamental question was found by Galileo and 
still more concisely by Newton, and forms a further 
clue in our investigation. 

To find the correct answer we must think a little 
more deeply about the cart on a perfectly smooth road. 


In our idealized experiment the uniformity of the mo- 
tion was due to the absence of all external forces. Let us 
now imagine that the uniformly moving cart is given 
a push in the direction of the motion. What happens 
now? Obviously its speed is increased. Just as ob- 
viously, a push in the direction opposite to that of the 
motion would decrease the speed. In the first case the 
cart is accelerated by the push, in the second case 
decelerated, or slowed down. A conclusion follows at 
once: the action of an external force changes the velo- 
city. Thus not the velocity itself but its change is a 
consequence of pushing or pulling. Such a force either 
increases or decreases the velocity according to whether 
it acts in the direction of motion or in the opposite 
direction. Galileo saw this clearly and wrote in his 
Two New Sciences : 

. . . any velocity once imparted to a moving body will 
be rigidly maintained as long as the external causes of 
acceleration or retardation are removed, a condition which 
is found only on horizontal planes; for in the case of 
planes which slope downwards there is already present a 
cause of acceleration; while on planes sloping upwards there 
is retardation; from this it follows that motion along a 
horizontal plane is perpetual; for, if the velocity be uni- 
form, it cannot be diminished or slackened, much less 

By following the right clue we achieve a deeper 
understanding of the problem of motion. The connec- 
tion between force and the change of velocity and 
not, as we should think according to our intuition, the 
connection between force and the velocity itself is 


the basis of classical mechanics as formulated by 

We have been making use of two concepts which 
play principal roles in classical mechanics: force and 
change of velocity. In the further development of 
science both of these concepts are extended and gener- 
alized. They must, therefore, be examined more closely. 

What is force? Intuitively, we feel what is meant by 
this term. The concept arose from the effort of push- 
ing, throwing or pulling from the muscular sensation 
accompanying each of these acts. But its generalization 
goes far beyond these simple examples. We can think 
of force even without picturing a horse pulling a car- 
riage ! We speak of the force of attraction between 
the sun and the earth, the earth and the moon, and of 
those forces which cause the tides. We speak of the 
force by which the earth compels ourselves and all the 
objects about us to remain within its sphere of in- 
fluence, and of the force with which the wind makes 
waves on the sea, or moves the leaves of trees. When 
and where we observe a change in velocity, an external 
force, in the general sense, must be held responsible. 
Newton wrote in his Principle, : 

An impressed force is an action exerted upon a body, 
in order to change its state, either of rest, or of moving 
uniformly forward in a right line. 

This force consists in the action only; and remains no 
longer in the body, when the action is over. For a body 
maintains every new state it acquires, by its vis inertiae 
only. Impressed forces are of different origins; as from 
percussion, from pressure, from centripetal force. 


If a stone is dropped from the top of a tower its mo- 
tion is by no means uniform; the velocity increases as 
the stone falls. We conclude: an external force is act- 
ing in the direction of the motion. Or, in other words : 
the earth attracts the stone. Let us take another ex- 
ample. What happens when a stone is thrown straight 
upward? The velocity decreases until the stone reaches 
its highest point and begins to fall. This decrease in 
velocity is caused by the same force as the acceleration 
of a falling body. In one case the force acts in the 
direction of the motion, in the other case in the opposite 
direction. The force is the same, but it causes accelera- 
tion or deceleration according to whether the stone is 
dropped or thrown upward. 


All motions we have been considering are rectilinear, 
that is, along a straight line. Now we must go one step 
farther. We gain an understanding of the laws of nature 
by analysing the simplest cases and by leaving out 
of our first attempts all intricate complications. A 
straight line is simpler than a curve. It is, however, 
impossible to be satisfied with an understanding of 
rectilinear motion alone. The motions of the moon, the 
earth and the planets, just those to which the principles 
of mechanics have been applied with such brilliant 
success, are motions along curved paths. Passing from 
rectilinear motion to motion along a curved path brings 
new difficulties. We must have the courage to over- 
come them if we wish to understand the principles 


of classical mechanics which gave us the first clues 
and so formed the starting-point for the development 
of science. 

Let us consider another idealized experiment, in 
which a perfect sphere rolls uniformly on a smooth 
table. We know that if the sphere is given a push, that 
is, if an external force is applied, the velocity will be 
changed. Now suppose that the direction of the blow 
is not, as in the example of the cart, in the line of 
motion, but in a quite different direction, say, perpen- 
dicular to that line. What happens to the sphere? 
Three stages of the motion can be distinguished: the 
initial motion, the action of the force, and the final 
motion after the force has ceased to act. According to 
the law of inertia, the velocities before and after the 
action of the force are both perfectly uniform. But 
there is a difference between the uniform motion be- 
fore and after the action of the force : the direction is 
changed. The initial path of the sphere and the direc- 
tion of the force are perpendicular to each other. The 
final motion will be along neither of these two lines, 
but somewhere between them, nearer the direction of 
the force if the blow is a hard one and the initial velo- 
city small, nearer the original line of motion if the blow 
is gentle and the initial velocity great. Our new con- 
clusion, based on the law of inertia, is : in general the 
action of an external force changes not only the speed 
but also the direction of the motion. An understanding 
of this fact prepares us for the generalization introduced 
into physics by the concept of vectors. 


We can continue to use our straightforward method 
of reasoning. The starting-point is again Galileo's law 
of inertia. We are still far from exhausting the conse- 
quences of this valuable clue to the puzzle of motion. 

Let us consider two spheres moving in different di- 
rections on a smooth table. So as to have a definite 
picture, we may assume the two directions perpendicu- 
lar to each other. Since there are no external forces 
acting, the motions are perfectly uniform. Suppose, 
further, that the speeds are equal, that is, both cover 
the same distance in the same interval of time. But 
is it correct to say that the two spheres have the same 
velocity? The answer can be yes or no ! If the speedo- 
meters of two cars both show forty miles per hour, it 
is usual to say that they have the same speed or velocity, 
no matter in which direction they are travelling. But 
science must create its own language, its own con- 
cepts, for its own use. Scientific concepts often begin 
with those used in ordinary language for the affairs, 
of everyday life, but they develop quite differently. 
They are transformed and lose the ambiguity associ- 
ated with them in ordinary language, gaining in rigor- 
ousness so that they may be applied to scientific 

From the physicist's point of view it is advantageous 
to say that the velocities of the two spheres moving in 
different directions are different. Although purely a 
matter of convention, it is more convenient to say that 
four cars travelling away from the same traffic round- 
about on different roads do not have the same velocity 


even though the speeds, as registered on the speedometers, 
are all forty miles per hour. This differentiation between 
speed and velocity illustrates how physics, starting with a 
concept used in everyday life, changes it in a way which 
proves fruitful in the further development of science. 

If a length is measured, the result is expressed as a 
number of units. The length of a stick may be 3 ft. 7 in. ; 
the weight of some object 2 Ib. 3 oz.; a measured 
time interval so many minutes or seconds. In each of 
these cases the result of the measurement is expressed 
by a number. A number alone is, however, insuffi- 
cient for describing some physical concepts. The re- 
cognition of this fact marked a distinct advance in 
scientific investigation. A direction as well as a number 
is essential for the characterization of a velocity, for 
example. Such a quantity, possessing both magnitude 
and direction, is called a vector. A suitable symbol for 


it is an arrow. Velocity may be represented by an 
arrow or, briefly speaking, by a vector whose length 
in some chosen scale of units is a measure of the speed, 
and whose direction is that of the motion. 

If four cars diverge with equal speed from a traffic 
roundabout, their velocities can be represented by four 
vectors of the same length, as seen from our last drawing. 
In the scale used, one inch stands for 40 m.p.h. In this 
way any velocity may be denoted by a vector, and 
conversely, if the scale is known, one may ascertain the 
velocity from such a vector diagram. 

If two cars pass each other on the highway and their 
speedometers both show 40 m.p.h., we characterize their 
velocities by two different vectors with arrows pointing 
in opposite directions. So also the arrows indicating 
"uptown" and "downtown" subway trains in New 

York must point in opposite directions. But all trains 
moving uptown at different stations or on different 
avenues with the same speed have the same velocity, 
which may be represented by a single vector. There is 
nothing about a vector to indicate which stations the 
train passes or on which of the many parallel tracks it 
is running. In other words, according to the accepted 
convention, all such vectors, as drawn below, may be 
regarded as equal; they lie along the same or parallel 
lines, are of equal length, and finally, have arrows 


pointing in the same direction. The next figure shows 
vectors all different, because they differ either in length 

or direction, or both. The same four vectors may be 
drawn in another way, in which they all diverge from 



a common point. Since the starting-point does not 
matter, these vectors can represent the velocities of four 
cars moving away from the same traffic roundabout, 
or the velocities of four cars in different parts of the 
country travelling with the indicated speeds in the 
indicated directions. 

This vector representation may now be used to de- 
scribe the facts previously discussed concerning recti- 
linear motion. We talked of a cart, moving uniformly 
in a straight line and receiving a push in the direction 
of its motion which increases its velocity. Graphically 
this may be represented by two vectors, a shorter one 
denoting the velocity before the push and a longer one 
in the same direction denoting the velocity after the 

push. The meaning of the dotted vector is clear; it 
represents the change in velocity for which, as we know, 
the push is responsible. For the case where the force 
is directed against the motion, where the motion is 
slowed down, the diagram is somewhat different. 


Again the dotted vector corresponds to a change in 
velocity, but in this case its direction is different. It is 
clear that not only velocities themselves but also their 
changes are vectors. But every change in velocity is 


due to the action of an external force; thus the force 
must also be represented by a vector. In order to 
characterize a force it is not sufficient to state how 
hard we push the cart; we must also say in which direc- 
tion we push. The force, like the velocity or its change, 
must be represented by a vector and not by a number 
alone. Therefore : the external force is also a vector, and 
must have the same direction as the change in velocity. 
In the two drawings the dotted vectors show the direction 
of the force as truly as they indicate the change in 

Here the sceptic may remark that he sees no ad- 
vantage in the introduction of vectors. All that has 
been accomplished is the translation of previously re- 
cognized facts into an unfamiliar and complicated 
language. At this stage it would indeed be difficult 
to convince him that he is wrong. For the moment he 
is, in fact, right. But we shall see that just this strange 
language leads to an important generalization in which 
vectors appear to be essential. 


So long as we deal only with motion along a straight 
line, we are far from understanding the motions ob- 
served in nature. We must consider motions along 
curved paths, and our next step is to determine the laws 
governing such motions. This is no easy task. In the 
case of rectilinear motion our concepts of velocity, 
change of velocity, and force proved most useful. But 
we do not immediately see how we can apply them to 


motion along a curved path. It is indeed possible to 
imagine that the old concepts are unsuited to the de- 
scription of general motion, and that new ones must 
be created. Should we try to follow our old path, or 
seek a new one? 

The generalization of a concept is a process very 
often used in science. A method of generalization is 
not uniquely determined, for there are usually numer- 
ous ways of carrying it out. One requirement, however, 
must be rigorously satisfied: any generalized concept 
must reduce to the original one when the original con- 
ditions are fulfilled. 

We can best explain this by the example with which 
we are now dealing. We can try to generalize the old 
concepts of velocity, change of velocity, and force for 
the case of motion along a curved path. Technically, 
when speaking of curves, we include straight lines. 
The straight line is a special and trivial example of a 
curve. If, therefore, velocity, change in velocity, and 
force are introduced for motion along a curved line, 
then they are automatically introduced for motion 
along a straight line. But this result must not contra- 
dict those results previously obtained. If the curve be- 
comes a straight line, all the generalized concepts must 
reduce to the familiar ones describing rectilinear mo- 
tion. But this restriction is not sufficient to determine 
the generalization uniquely. It leaves open many pos- 
sibilities. The history of science shows that the simplest 
generalizations sometimes prove successful and some- 
times not. We must first make a guess. In our case it is 


a simple matter to guess the right method of generaliza- 
tion. The new concepts prove very successful and help 
us to understand the motion of a thrown stone as well 
as that of the planets. 

And now just what do the words velocity, change in 
velocity, and force mean in the general case of motion 
along a curved line? Let us begin with velocity. Along 
the curve a very small body is moving from left to 

right. Such a small body is often called a particle. The 
dot on the curve in our drawing shows the position of 
the particle at some instant of time. What is the velocity 
corresponding to this time and position? Again Galileo's 
clue hints at a way of introducing the velocity. We must, 
once more, use our imagination and think about an 
idealized experiment. The particle moves along the 
curve, from left to right, under the influence of ex- 
ternal forces. Imagine that at a given time, and at 
the point indicated by the dot, all these forces suddenly 
cease to act. Then, the motion must, according to the 
law of inertia, be uniform. In practice we can, of 
course, never completely free a body from all external 
influences. We can only surmise "what would happen 
if. . . ?" and judge the pertinence of our guess by the 
conclusions which can be drawn from it and by their 
agreement with experiment. 



The vector in the next drawing indicates the guessed 
direction of the uniform motion if all external forces 

were to vanish. It is the direction of the so-called 
tangent. Looking at a moving particle through a 
microscope one sees a very small part of the curve, 
which appears as a small segment. The tangent is its 
prolongation. Thus the vector drawn represents the 
velocity at a given instant. The velocity vector lies on 
the tangent. Its length represents the magnitude of the 
velocity, or the speed as indicated, for instance, by the 
speedometer of a car. 

Our idealized experiment about destroying the mo- 
tion in order to find the velocity vector must not be 
taken too seriously. It only helps us to understand what 
we should call the velocity vector and enables us to 
determine it for a given instant at a given point. 

In the next drawing, the velocity vectors for three 
different positions of a particle moving along a curve 

are shown. In this case not only the direction but the 
magnitude of the velocity, as indicated by the length 
of the vector, varies during the motion. 


Does this new concept of velocity satisfy the require- 
ment formulated for all generalizations? That is: does 
it reduce to the familiar concept if the curve becomes 
a straight line? Obviously it does. The tangent to a 
straight line is the line itself. The velocity vector lies in 
the line of the motion, just as in the case of the moving 
cart or the rolling spheres. 

The next step is the introduction of the change in 
velocity of a particle moving along a curve. This also 
may be done in various ways, from which we choose 
the simplest and most convenient. The last drawing 
showed several velocity vectors representing the mo- 
tion at various points along the path. The first two of 
these may be drawn again so that they have a common 
starting-point, as we have seen is possible with vectors. 

The dotted vector we call the change in velocity. Its 
starting-point is the end of the first vector and its end- 
point the end of the second vector. This definition of 
the change in velocity may, at first sight, seem artificial 
and meaningless. It becomes much clearer in the special 
case in which vectors (i) and (2) have the same direc- 
tion. This, of course, means going over to the case of 
straight-line motion. If both vectors have the same 
initial point, the dotted vector again connects their end- 
points. The drawing is now identical with that on p. 18, 


*-> ......... > 

and the previous concept is regained as a special 
case of the new one. We may remark that we had to 
separate the two lines in our drawing, since otherwise 
they would coincide and be indistinguishable. 

We now have to take the last step in our process of 
generalization. It is the most important of all the guesses 
we have had to make so far. The connection between 
force and change in velocity has to be established so 
that we can formulate the clue which will enable us to 
understand the general problem of motion. 

The clue to an explanation of motion along a straight 
line was simple : external force is responsible for change 
in velocity; the force vector has the same direction as 
the change. And now what is to be regarded as the 
clue to curvilinear motion? Exactly the same ! The 
only difference is that change of velocity has now a 
broader meaning than before. A glance at the dotted 
vectors of the last two drawings shows this point clearly. 
If the velocity is known for all points along the curve, 
the direction of the force at any point can be deduced 
at once. One must draw the velocity vectors for two 
instants separated by a very short time interval and 
therefore corresponding to positions very near each 
other. The vector from the end-point of the first to 
that of the second indicates the direction of the acting 
force. But v it is essential that the two velocity vectors 


should be separated only by a "very short" time 
interval. The rigorous analysis of such words as "very 
near", "very short" is far from simple. Indeed it was 
this analysis which led Newton and Leibnitz to the 
discovery of differential calculus. 

It is a tedious and elaborate path which leads to the 
generalization of Galileo's clue. We cannot show here 
how abundant and fruitful the consequences of this 
generalization have proved. Its application leads to 
simple and convincing explanations of many facts pre- 
viously incoherent and misunderstood. 

From the extremely rich variety of motions we shall 
take only the simplest and apply to their explanation 
the law just formulated. 

A bullet shot from a gun, a stone thrown at an angle, 
a stream of water emerging from a hose, all describe 
familiar paths of the same type the parabola. Imagine 
a speedometer attached to a stone, for example, so 
that its velocity vector may be drawn for any instant. 

The result may well be that represented in the above 
drawing. The direction of the force acting on the stone 
is just that of the change in velocity, and we have seen 
how it may be determined. The result, shown in the 
next drawing, indicates that the force is vertical, and 



directed downward. It is exactly the same as when a 
stone is allowed to fall from the top of a tower. The 
paths are quite different, as also are the velocities, but 
the change in velocity has the same direction, that is, 
toward the centre of the earth. 

A stone attached to the end of a string and swung 
around in a horizontal plane moves in a circular path. 

All the vectors in the diagram representing this mo- 
tion have the same length if the speed is uniform. 

Nevertheless, the velocity is not uniform, for the path 
is not a straight line. Only in uniform, rectilinear mo- 


tion are there no forces involved. Here, however, there 
are, and the velocity changes not in magnitude but in 
direction. According to the law of motion there must 
be some force responsible for this change, a force in 
this case between the stone and the hand holding the 
string. A further question arises immediately: in what 
direction does the force act? Again a vector diagram 
shows the answer. The velocity vectors for two very 
near points are drawn, and the change of velocity 

found. This last vector is seen to be directed along the 
string toward the centre of the circle, and is always 
perpendicular to the velocity vector, or tangent. In 
other words, the hand exerts a force on the stone by 
means of the string. 

Very similar is the more important example of the 
revolution of the moon around the earth. This may be 
represented approximately as uniform circular mo- 
tion. The force is directed toward the earth for the 
same reason that it was directed toward the hand in 
our former example. There is no string connecting the 
earth and the moon, but we can imagine a line be- 
tween the centres of the two bodies; the force lies 
along this line and is directed toward the centre of the 
earth, just as the force on a stone thrown in the air or 
dropped from a tower. 


All that we have said concerning motion can be 
summed up in a single sentence. Force and change of 
velocity are vectors having the same direction. This is the 
initial clue to the problem of motion, but it certainly 
does not suffice for a thorough explanation of all mo- 
tions observed. The transition from Aristotle's line of 
thought to that of Galileo formed a most important 
corner-stone in the foundation of science. Once this 
break was made, the line of further development was 
clear. Our interest here lies in the first stages of develop- 
ment, in following initial clues, in showing how new 
physical concepts are born in the painful struggle with 
old ideas. We are concerned only with pioneer work 
in science, which consists of finding new and unex- 
pected paths of development ; with the adventures in 
scientific thought which create an ever-changing pic- 
ture of the universe. The initial and fundamental steps 
are always of a revolutionary character. Scientific ima- 
gination finds old concepts too confining, and replaces 
them by new ones. The continued development along 
any line already initiated is more in the nature of evo- 
lution, until the next turning point is reached when a 
still newer field must be conquered. In order to under- 
stand, however, what reasons and what difficulties force 
a change in important concepts, we must know not only 
the initial clues, but also the conclusions which can be 

One of the most important characteristics of modern 
physics is that the conclusions drawn from initial clues 
are not only qualitative but also quantitative. Let us 


again consider a stone dropped from a tower. We have 
seen that its velocity increases as it falls, but we should 
like to know much more. Just how great is this change? 
And what is the position and the velocity of the stone 
at any time after it begins to fall? We wish to be able 
to predict events and to determine by experiment 
whether observation confirms these predictions and 
thus the initial assumptions. 

To draw quantitative conclusions we must use the 
language of mathematics. Most of the fundamental 
ideas of science are essentially simple, and may, as a 
rule, be expressed in a language comprehensible to 
everyone. To follow up these ideas demands the know- 
ledge of a highly refined technique of investigation. 
Mathematics as a tool of reasoning is necessary if we 
wish to draw conclusions which may be compared 
with experiment. So long as we are concerned only 
with fundamental physical ideas we may avoid the 
language of mathematics. Since in these pages we do 
this consistently, we must occasionally restrict our- 
selves to quoting, without proof, some of the results 
necessary for an understanding of important clues 
arising in the further development. The price which 
must be paid for abandoning the language of mathe- 
matics is a loss in precision, and the necessity of some- 
times quoting results without showing how they were 

A very important example of motion is that of the 
earth around the sun. It is known that the path is a 
closed curve, called the ellipse. The construction of a 


vector diagram of the change in velocity shows that 
the force on the earth is directed toward the sun. But 


this, after all, is scant information. We should like to 
be able to predict the position of the earth and the 
other planets for any arbitrary instant of time, we 
should like to predict the date and duration of the 
next solar eclipse and many other astronomical events. 
It is possible to do these things, but not on the basis of 
our initial clue alone, for it is now necessary to know 
not only the direction of the force but also its absolute 
value its magnitude. It was Newton who made the 
inspired guess on this point. According to his law of 
gravitation the force of attraction between two bodies 
depends in a simple way on their distance from each 
other. It becomes smaller when the distance increases. 
To be specific it becomes 2x2=4 times smaller if 
the distance is doubled, 3x3 = 9 times smaller if the 
distance is made three times as great. 

Thus we see that in the case of gravitational force 
we have succeeded in expressing, in a simple way, the 


dependence of the force on the distance between the 
moving bodies. We proceed similarly in all other cases 
where forces of different kinds for instance, electric, 
magnetic, and the like are acting. We try to use a 
simple expression for the force. Such an expression is 
justified only when the conclusions drawn from it are 
confirmed by experiment. 

But this knowledge of the gravitational force alone 
is not sufficient for a description of the motion of the 
planets. We have seen that vectors representing force 
and change in velocity for any short interval of time 
have the same direction, but we must follow Newton 
one step farther and assume a simple relation between 
their lengths. Given all other conditions the same, that 
is, the same moving body and changes considered over 
equal time intervals, then, according to Newton, the 
change of velocity is proportional to the force. 

Thus just two complementary guesses are needed 
for quantitative conclusions concerning the motion of 
the planets. One is of a general character, stating the 
connection between force and change in velocity. The 
other is special, and states the exact dependence of the 
particular kind of force involved on the distance be- 
tween the bodies. The first is Newton's general law of 
motion, the second his law of gravitation. Together 
they determine the motion. This can be made clear by 
the following somewhat clumsy-sounding reasoning. 
Suppose that at a given time the position and velocity 
of a planet can be determined, and that the force is 
known. Then, according to Newton's laws, we know 


the change in velocity during a short time interval. 
Knowing the initial velocity and its change, we can 
find the velocity and position of the planet at the end 
of the time interval. By a continued repetition of this 
process the whole path of the motion may be traced 
without further recourse to observational data. This is, 
in principle, the way mechanics predicts the course of 
a body in motion, but the method used here is hardly 
practical. In practice such a step-by-step procedure 
would be extremely tedious as well as inaccurate. For- 
tunately, it is quite unnecessary; mathematics furnishes 
a short cut, and makes possible precise description of 
the motion in much less ink than we use for a single 
sentence. The conclusions reached in this way can be 
proved or disproved by observation. 

The same kind of external force is recognized in the 
motion of a stone falling through the air and in the 
revolution of the moon in its orbit, namely, that of the 
earth's attraction for material bodies. Newton recog- 
nized that the motions of falling stones, of the moon, 
and of planets are only very special manifestations of a 
universal gravitational force acting between any two 
bodies. In simple cases the motion may be described 
and predicted by the aid of mathematics. In remote 
and extremely complicated cases, involving the action 
of many bodies on each other, a mathematical descrip- 
tion is not so simple, but the fundamental principles 
are the same. 

We find the conclusions, at which we arrived by 
following our initial clues, realized in the motion of a 


thrown stone, in the motion of the moon, the earth, 
and the planets. 

It is really our whole system of guesses which is to 
be either proved or disproved by experiment. No one 
of the assumptions can be isolated for separate testing. 
In the case of the planets moving around the sun it is 
found that the system of mechanics works splendidly. 
Nevertheless we can well imagine that another system, 
based on different assumptions, might work just as 

Physical concepts are free creations of the human 
mind, and are not, however it may seem, uniquely de- 
termined by the external world. In our endeavour to 
understand reality we are somewhat like a man trying 
to understand the mechanism of a closed watch. He 
sees the face and the moving hands, even hears its tick- 
ing, but he has no way of opening the case. If he is 
ingenious he may form some picture of a mechanism 
which could be responsible for all the things he ob- 
serves, but he may never be quite sure his picture is the 
only one which could explain his observations. He will 
never be able to compare his picture with the real 
mechanism and he cannot even imagine the possibility 
or the meaning of such a comparison. But he certainly 
believes that, as his knowledge increases, his picture of 
reality will become simpler and simpler and will explain 
a wider and wider range of his sensuous impressions. 
He may also believe in the existence of the ideal limit 
of knowledge and that it is approached by the human 
mind. He may call this ideal limit the objective truth. 



When first studying mechanics one has the impres- 
sion that everything in this branch of science is simple, 
fundamental and settled for all time. One would hardly 
suspect the existence of an important clue which no 
one noticed for three hundred years. The neglected 
clue is connected with one of the fundamental concepts 
of mechanics that of mass. 

Again we return to the simple idealized experiment 
of the cart on a perfectly smooth road. If the cart is 
initially at rest and then given a push, it afterwards 
moves uniformly with a certain velocity. Suppose that 
the action of the force can be repeated as many times 
as desired, the mechanism of pushing acting in the 
same way and exerting the same force on the same 
cart. However many times the experiment is repeated, 
the final velocity is always the same. But what happens 
if the experiment is changed, if previously the cart was 
empty and now it is loaded? The loaded cart will have 
a smaller final velocity than the empty one. The con- 
clusion is: if the same force acts on two different 
bodies, both initially at rest, the resulting velocities 
will not be the same. We say that the velocity depends 
on the mass of the body, being smaller if the mass is 

We know, therefore, at least in theory, how to de- 
termine the mass of a body or, more exactly, how 
many times greater one mass is than another. We have 
identical forces acting on two resting masses. Finding 


that the velocity of the first mass is three times greater 
than that of the second, we conclude that the first mass 
is three times smaller than the second. This is certainly 
not a very practical way of determining the ratio of 
two masses. We can, nevertheless, well imagine having 
done it in this, or in some similar way, based upon the 
application of the law of inertia. 

How do we really determine mass in practice? Not, 
of course, in the way just described. Everyone knows 
the correct answer. We do it by weighing on a scale. 

Let us discuss in more detail the two different ways 
of determining mass. 

The first experiment had nothing whatever to do 
with gravity, the attraction of the earth. The cart 
moves along a perfectly smooth and horizontal plane 
after the push. Gravitational force, which causes the 
cart to stay on the plane, does not change, and plays no 
role in the determination of the mass. It is quite differ- 
ent with weighing. We could never use a scale if the 
earth did not attract bodies, if gravity did not exist. 
The difference between the two determinations of 
mass is that the first has nothing to do with the force 
of gravity while the second is based essentially on its 

We ask : if we determine the ratio of two masses in 
both ways described above, do we obtain the same re- 
sult? The answer given by experiment is quite clear. 
The results are exactly the same ! This conclusion could 
not have been foreseen, and is based on observation, 
not reason. Let us, for the sake of simplicity, call the 


mass determined in the first way the inertial mass and 
that determined in the second way the gravitational 
mass. In our world it happens that they are equal, but 
we can well imagine that this should not have been the 
case at all. Another question arises immediately : is this 
identity of the two kinds of mass purely accidental, or 
does it have a deeper significance? The answer, from 
the point of view of classical physics, is : the identity of 
the two masses is accidental and no deeper significance 
should be attached to it. The answer of modern physics 
is just the opposite: the identity of the two masses is 
fundamental and forms a new and essential clue lead- 
ing to a more profound understanding. This was, in 
fact, one of the most important clues from which the 
so-called general theory of relativity was developed. 

A mystery story seems inferior if it explains strange 
events as accidents. It is certainly more satisfying to 
have the story follow a rational pattern. In exactly the 
same way a theory which offers an explanation for the 
identity of gravitational and inertial mass is superior 
to one which interprets their identity as accidental, 
provided, of course, that the two theories are equally 
consistent with observed facts. 

Since this identity of inertial and gravitational mass 
was fundamental for the formulation of the theory of 
relativity, we are justified in examining it a little more 
closely here. What experiments prove convincingly 
that the two masses are the same? The answer lies in 
Galileo's old experiment in which he dropped different 
masses from a tower. He noticed that the time required 


for the fall was always the same, that the motion of a 
falling body does not depend on the mass. To link this 
simple but highly important experimental result with 
the identity of the two masses needs some rather in- 
tricate reasoning. 

A body at rest gives way before the action of an ex- 
ternal force, moving and attaining a certain velocity. It 
yields more or less easily, according to its inertial mass, 
resisting the motion more strongly if the mass is large 
than if it is small. We may say, without pretending to 
be rigorous: the readiness with which a body responds 
to the call of an external force depends on its inertial 
mass. If it were true that the earth attracts all bodies 
with the same force, that of greatest inertial mass 
would move more slowly in falling than any other. But 
this is not the case: all bodies fall in the same way. 
This means that the force by which the earth attracts 
different masses must be different. Now the earth at- 
tracts a stone with the force of gravity and knows 
nothing about its inertial mass. The "calling" force of 
the earth depends on the gravitational mass. The " an- 
swering" motion of the stone depends on the inertial 
mass. Since the " answering " motion is always the same 
all bodies dropped from the same height fall in the 
same way it must be deduced that gravitational mass 
and inertial mass are equal. 

More pedantically a physicist formulates the same 
conclusion : the acceleration of a falling body increases 
in proportion to its gravitational mass and decreases in 
proportion to its inertial mass. Since all falling bodies 


have the same constant acceleration, the two masses 
must be equal. 

In our great mystery story there are no problems 
wholly solved and settled for all time. After three hun- 
dred years we had to return to the initial problem of 
motion, to revise the procedure of investigation, to 
find clues which had been overlooked, thereby reach- 
ing a different picture of the surrounding universe. 


Here we begin to follow a new clue, one originating in 
the realm of heat phenomena. It is impossible, however, 
to divide science into separate and unrelated sections. 
Indeed, we shall soon find that the new concepts intro- 
duced here are interwoven with those already familiar, 
and with those we shall still meet. A line of thought 
developed in one branch of science can very often be 
applied to the description of events apparently quite 
different in character. In this process the original 
concepts are often modified so as to advance the under- 
standing both of those phenomena from which they 
sprang and of those to which they are newly applied. 

The most fundamental concepts in the description 
of heat phenomena are temperature and heat. It took 
an unbelievably long time in the history of science for 
these two to be distinguished, but once this distinction 
was made rapid progress resulted. Although these con- 
cepts are now familiar to everyone, we shall examine 
them closely, emphasizing the differences between 


Our sense of touch tells us quite definitely that one 
body is hot and another cold. But this is a purely quali- 
tative criterion, not sufficient for a quantitative descrip- 
tion and sometimes even ambiguous. This is shown by 
a well-known experiment: we have three vessels con- 
taining, respectively, cold, warm and hot water. If we 
dip one hand into the cold water and the other into the 
hot, we receive a message from the first that it is cold 
and from the second that it is hot. If we then dip both 
hands into the same warm water, we receive two con- 
tradictory messages, one from each hand. For the same 
reason an Eskimo and a native of some equatorial coun- 
try meeting in New York on a spring day would hold 
different opinions as to whether the climate was hot 
or cold. We settle all such questions by the use of a 
thermometer, an instrument designed in a primitive 
form by Galileo. Here again that familiar name ! The 
use of a thermometer is based on some obvious physical 
assumptions. We shall recall them by quoting a few 
lines from lectures given about a hundred and fifty 
years ago by Black, who contributed a great deal 
toward clearing up the difficulties connected with the 
two concepts, heat and temperature : 

By the use of this instrument we have learned, that if we 
take 1000, or more, different kinds of matter, such as metals, 
stones, salts, woods, feathers, wool, water and a variety 
of other fluids, although they be all at first of different 
heats, let them be placed together in the same room without 
a fire, and into which the sun does not shine, the heat will 
be communicated from the hotter of these bodies to the 
colder, during some hours perhaps, or the course of a 


at the end of which time, if we apply a thermometer to 
them all in succession, it will point precisely to the same 

The italicized word heats should, according to present- 
day nomenclature, be replaced by the word tempera- 

A physician taking the thermometer from a sick 
man's mouth might reason like this: "The thermometer 
indicates its own temperature by the length of its col- 
umn of mercury. We assume that the length of the 
mercury column increases in proportion to the increase 
in temperature. But the thermometer was for a few 
minutes in contact with my P^ientjjojhat bothjjatient 
and thermometer havef the Tame temperature. I con- 
cIuHe, therefore, that my patient's temperature is that 
registered on the thermometer." The doctor probably 
acts mechanically, but he applies physical principles 
without thinking about it. 

But does the thermometer contain the same amount 
of heat as the body of the man? Of course not. To 
assume that two bodies contain equal quantities of heat 
just because their temperatures are equal would, as 
Black remarked, be 

taking a very hasty view of the subject. It is confounding 
the quantity of heat in different bodies with its general 
strength or intensity, though it is plain that these are two 
different things, and should always be distinguished, when 
we are thinking of the distribution of heat. 

An understanding of this distinction can be gained 
by considering a very simple experiment. A pound of 


water placed over a gas flame takes some time to 
change from room temperature to the boiling point. 
A much longer time is required for heating twelve 
pounds, say, of water in the same vessel by means of 
the same flame. We interpret this fact as indicating that 
now more of "something" is needed and we call this 
"something" heat. 

A further important concept, specific heat^is^ gained 
by the following experiment: let one vessel contain a 
pound of water and another a pound of mercury, both 
to be heated in the same way. The mercury gets hot 
much more quickly than the water, showing that less 
"heat" is needed to raise the temperature by one degree. 
In general ? different amounts of "heat" are required to 
change by one degree, say from ^ojtp 41 degrees Fah- 
renheit, the temperatures of different substances such 
as~water, mercury, iron, copper, woodTelc^ aTroTtHc 
samelrTass. We say that each substance has its individual 
heat capacit^^j)Y, specific heat. 

Once having gained the concept of heat, we can in- 
vestigate its nature more closely. We have two bodies, 
one hot, the other cold, or more precisely, one of a 
higher temperature than the other. We bring them into 
contact and free them from all other external in- 
fluences. Eventually they will, we know, reach the 
same temperature. But how does this take place? What 
happens between the instant they are brought into 
contact and the achievement of equal temperatures? 
The picture of heat "flowing" from one body to an- 
other suggests itself, like water flowing from a higher 


level to a lower. This picture, though primitive, seems 
to fit many of the facts, so that the analogy runs : 

Water Heat 

Higher level Higher temperature 
Lower level Lower temperature 

The flow proceeds until both levels, that is, both tem- 
peratures, are equal. This naive view can be made more 
useful by quantitative considerations. If definite masses 
of water and alcohol, each at a definite temperature, 
are mixed together, a knowledge of the specific heats 
will lead to a prediction of the final temperature of the 
mixture. Conversely, an observation of the final tem- 
perature, together with a little algebra, would enable 
us to find the ratio of the two specific heats. 

We recognize in the concept of heat which appears 
here a similarity to other physical concepts. Heat is, 
according to our view, a substance, such as mass in 
mechanics. Its quantity may change or not, like money 
put aside in a safe or spent. The amount of money in a 
safe will remain unchanged so long as the safe remains 
locked, and so will the amounts of mass and heat in an 
isolated body. The ideal thermos flask is analogous to 
such a safe. Furthermore, just as the mass of an iso- 
lated system is unchanged even if a chemical trans- 
formation takes place, so heat is conserved even though 
it flows from one body to another. Even if heat is not 
used for raising the temperature of a body but for 
melting ice, say, or changing water into steam, we can 
still think of it as a substance and regain it entirely by 


freezing the water or liquefying the steam. The old 
names, latent heat of melting or vaporization, show 
that these concepts are drawn from the picture of heat 
as a substance. Latent heat is temporarily hidden, like 
money put away in a safe, but available for use if one 
knows the lock combination. 

But heat is certainly not a substance in the same sense 
as mass. Mass can be detected by means of scales, but 
what of heat? Does a piece of iron weigh more when 
red-hot than when ice-cold? Experiment shows that it 
does not. If heat is a substance at all, it is a weightless 
one. The "heat-substance" was usually called caloric 
and is our first acquaintance among a whole family of 
weightless substances. Later we shall have occasion to 
follow the history of the family, its rise and fall. It is 
sufficient now to note the birth of this particular mem- 

The purpose of any physical theory is to explain as 
wide a range of phenomena as possible. It is justified in 
so far as it does make events understandable. We have 
seen that the substance theory explains many of the 
heat phenomena. It will soon become apparent, how- 
ever, that this again is a false clue, that heat cannot be 
regarded as a substance, even weightless. This is clear 
if we think about some simple experiments which 
marked the beginning of civilization. 

We think of a substance as something which can 
be neither created nor destroyed. Yet primitive man 
created by friction sufficient heat to ignite wood. Ex- 
amples of heating by friction are, as a matter of fact, 


much too numerous and familiar to need recounting. 
In all these cases some quantity of heat is created, a 
fact difficult to account for by the substance theory. It 
is true that a supporter of this theory could invent 
arguments to account for it. His reasoning would run 
something like this: u The substance theory can explain 
the apparent creation of heat. Take the simplest ex- 
ample of two pieces of wood rubbed one against the 
other. Now rubbing is something which influences the 
wood and changes its properties. It is very likely that 
the properties are so modified that an unchanged quan- 
tity of heat comes to produce a higher temperature than 
before. After all, the only thing we notice is the rise 
in temperature. It is possible that the friction changes 
the specific heat of the wood and not the total amount 
of heat." 

At this stage of the discussion it would be useless to 
argue with a supporter of the substance theory, for this 
is a matter which can be settled only by experiment. 
Imagine two identical pieces of wood and suppose 
equal changes of temperature are induced by different 
methods; in one case by friction and in the other by 
contact with a radiator, for example. If the two pieces 
have the same specific heat at the new temperature, the 
whole substance theory must break down. There are very 
simple methods for determining specific heats, and the 
fate of the theory depends on the result of just such 
measurements. Tests which are capable of pronounc- 
ing a verdict of life or death on a theory occur fre- 
quently in the history of physics, and are called crucial 


experiments. The crucial value of an experiment is re- 
vealed only by the way the question is formulated, and 
only one theory of the phenomena can be put on trial 
by it. The determination of the specific heats of two 
bodies of the same kind, at equal temperatures attained 
by friction and heat flow respectively, is a typical ex- 
ample of a crucial experiment. This experiment was 
performed about a hundred and fifty years ago by 
Rumford, and dealt a death blow to the substance 
theory of heat. 

An extract from Rurnford's own account tells the 

It frequently happens, that in the ordinary affairs and 
occupations of life, opportunities present themselves of con- 
templating some of the most curious operations of Nature; 
and very interesting philosophical experiments might often 
be made, almost without trouble or expense, by means of 
machinery contrived for the mere mechanical purposes of 
the arts and manufactures, 

I have frequently had occasion to make this observation; 
and am persuaded, that a habit of keeping the eyes open to 
every thing that is going on in the ordinary course of the 
business of life has oftener led, as it were by accident, or in 
the playful excursions of the imagination, put into action 
by contemplating the most common appearances, to useful 
doubts, and sensible schemes for investigation and improve- 
ment, than all the more intense meditations of philosophers, 
in the hours expressly set apart for study 

Being engaged, lately, in superintending the boring of 
cannon, in the workshops of the military arsenal at Munich, 
I was struck with the very considerable degree of Heat 
which a brass gun acquires, in a short time, in being bored ; 
and with the still more intense Heat (much greater than 


that of boiling water, as I found by experiment) of the 
metallic chips separated from it by the borer .... 

From whence comes the Heat actually produced in the 
mechanical operation above mentioned? 

Is it furnished by the metallic chips which are separated 
by the borer from the solid mass of metal? 

If this were the case, then, according to the modern 
doctrines of latent Heat, and of caloric, the capacity ought 
not only to be changed, but the change undergone by them 
should be sufficiently great to account for all the Heat 

But no such change had taken place; for I found, upon 
taking equal quantities, by weight, of these chips, and of 
thin slips of the same block of metal separated by means 
of a fine saw and putting them, at the same temperature 
(that of boiling water), into equal quantities of cold water 
(that is to say, at the temperature of 59^ F.) the portion 
of water into which the chips were put was not, to all 
appearance, heated either less or more than the other 
portion, in which the slips of metal were put. 

Finally we reach his conclusion : 

And, in reasoning on this subject, we must not forget to 
consider that most remarkable circumstance, that the source 
of the Heat generated by friction, in these Experiments, 
appeared evidently to be inexhaustible. 

It is hardly necessary to add, that anything which any 
insulated body, or system of bodies, can continue to furnish 
without limitation, cannot possibly be a material substance] and 
it appears to me to be extremely difficult, if not quite im- 
possible, to form any distinct idea of anything, capable of 
being excited and communicated, in the manner the Heat 
was excited and communicated in these Experiments, except 
it be MOTION. 

Thus we see the breakdown of the old theory, or to 
be more exact, we see that the substance theory is 


limited to problems of heat flow. Again, as Rumford 
has intimated, we must seek a new clue. To do this, 
let us leave for the moment the problem of heat and 
return to mechanics. 


Let us trace the motion of that popular thrill-giver, 
the switchback. A small car is lifted or driven to the 
highest point of the track. When set free it starts roll- 
ing down under the force of gravity, and then goes up 
and down along a fantastically curved line, giving the 
occupants a thrill by the sudden changes in velocity. 
Every switchback has its highest point, that from which 
it starts. Never again, throughout the whole course of 
the motion, will it reach the same height. A complete 
description of the motion would be very complicated. 
On the one hand is the mechanical side of the pro- 
blem, the changes of velocity and position in time. On 
the other there is friction and therefore the creation 
of heat, on the rail and in the wheels. The only signifi- 
cant reason for dividing the physical process into these 
two aspects is to make possible the use of the concepts 
previously discussed. The division leads to an idealized 
experiment, for a physical process in which only the 
mechanical aspect appears can be only imagined but 
never realized. 

For the idealized experiment we may imagine that 
someone has learned to eliminate entirely the friction 
which always accompanies motion. He decides to apply 
his discovery to the construction of a switchback, and 


must find out for himself how to build one. The car 
is to run up and down, with its starting-point, say, at 
one hundred feet above ground level. He soon dis- 
covers by trial and error that he must follow a very 
simple rule: he may build his track in whatever path 
he pleases so long as no point is higher than the starting- 
point. If the car is to proceed freely to the end of the 

course, its height may attain a hundred feet as many 
times as he likes, but never exceed it. The initial height 
can never be reached by a car on an actual track 
because of friction, but our hypothetical engineer need 
not consider that. 

Let us follow the motion of the idealized car on the 
idealized switchback as it begins to roll downward 
from the starting-point. As it moves its distance from 
the ground diminishes, but its speed increases. This 
sentence at first sight may remind us of one from a 
language lesson: "I have no pencil, but you have six 
oranges." It is not so stupid, however. There is no 
connection between my having no pencil and your 
having six oranges, but there is a very real correlation 


between the distance of the car from the ground and 
its speed. We can calculate the speed of the car at any 
moment if we know how high it happens to be above 
the ground, but we omit this point here because of its 
quantitative character which can best be expressed by 
mathematical formulae. 

At its highest point the car has zero velocity and is 
one hundred feet from the ground. At the lowest pos- 
sible point it is no distance from the ground, and has 
its greatest velocity. These facts may be expressed in 
other terms. At its highest point the car has potential 
energy but no kinetic energy or energy of motion. At its 
lowest point it has the greatest kinetic energy and no 
potential energy whatever. At all intermediate posi- 
tions, where there is some velocity and some elevation, 
it has both kinetic and potential energy. The potential 
energy increases with the elevation, while the kinetic 
energy becomes greater as the velocity increases. The 
principles of mechanics suffice to explain the motion. 
Two expressions for energy occur in the mathematical 
description, each of which changes, although the sum 
does not vary. It is thus possible to introduce mathema- 
tically and rigorously the concepts of potential energy, 
depending on position, and kinetic energy, depending 
on velocity. The introduction of the two names is, of 
course, arbitrary and justified only by convenience. 
The sum of the two quantities remains unchanged, and 
is called a constant of the motion. The total energy, 
kinetic plus potential, can be compared, for example, 
with money kept intact as to amount but changed 


continually from, one currency to another, say from 
dollars to pounds and back again, according to a well- 
defined rate of exchange. 

In the real switchback, where friction prevents the 
car from again reaching as high a point as that from 
which it started, there is still a continuous change be- 
tween kinetic and potential energy. Here, however, 
the sum does not remain constant, but grows smaller. 

Now one important and courageous step more is needed 
to relate the mechanical and heat aspects of motion. 
The wealth of consequences and generalizations from 
this step will be seen later. 

Something more than kinetic and potential energies 
is now involved, namely, the heat created by friction. 
Does this heat correspond to the diminution in me- 
chanical energy, that is kinetic and potential energy? 
A new guess is imminent. If heat may be regarded as a 
form of energy, perhaps the sum of all three heat, 
kinetic and potential energies remains constant. Not 
heat alone, but heat and other forms of energy taken 


together are, like a substance, indestructible. It is as if 
a man must pay himself a commission in francs for 
changing dollars to pounds, the commission money also 
being saved so that the sum of dollars, pounds, and 
francs is a fixed amount according to some defiYiite 
exchange rate. 

The progress of science has destroyed the older con- 
cept of heat as a substance. We try to create a new 
substance, energy, with heat as one of its forms. 


Less than a hundred years ago the new clue which 
led to the concept of heat as a form of energy was 
guessed by Mayer and confirmed experimentally by 
Joule. It is a strange coincidence that nearly all the 
fundamental work concerned with the nature of heat 
was done by non-professional physicists who regarded 
physics merely as their great hobby. There was the 
versatile Scotsman Black, the German physician Mayer, 
and the great American adventurer Count Rumford, 
who afterwards lived in Europe and, among other 
activities, became Minister of War for Bavaria. There 
was also the English brewer Joule who, in his spare 
time, performed some most important experiments con- 
cerning the conservation of energy. 

Joule verified by experiment the guess that heat is 
a form of energy, and determined the rate of exchange. 
It is worth our while to see just what his results were. 

The kinetic and potential energy of a system to- 
gether constitute its mechanical energy. In the case of 


the switchback we made a guess that some of the 
mechanical energy was converted into heat. If this is 
right, there must be here and in all other similar 
physical processes a definite rate of exchange between 
the two. This is rigorously a quantitative question, but 
the fact that a given quantity of mechanical energy 
can be changed into a definite amount of heat is highly 
important. We should like to know what number 
expresses the rate of exchange, i.e., how much 
heat we obtain from a given amount of mechanical 

The determination of this number was the object of 
Joule's researches. The mechanism of one of his experi- 
ments is very much like that of a weight clock. The 
winding of such a clock consists of elevating two weights, 
thereby adding potential energy to the system. If the 
clock is not further interfered with, it may be regarded 
as a closed system. Gradually the weights fall and the 
clock runs down. At the end of a certain time the 
weights will have reached their lowest position and 
the clock will have stopped. What has happened to 
the energy? The potential energy of the weights has 
changed into kinetic energy of the mechanism, and 
has then gradually been dissipated as heat. 

A clever alteration in this sort of mechanism enabled 
Joule to measure the heat lost and thus the rate of 
exchange. In his apparatus two weights caused a 
paddle wheel to turn while immersed in water. The 
potential energy of the weights was changed into kinetic 
energy of the movable parts, and thence into heat which 



raised the temperature of the water. Joule measured 
this change of temperature and, making use of the 
known specific heat of water, calculated the amount 
of heat absorbed. He summarized the results of many 
trials as follows : 

i st. That the quantity of heat produced by the friction of 
bodies, whether solid or liquid, is always proportional to the 
quantity of force [by force Joule means energy] expended. 

2nd. That the quantity of heat capable of increasing the 
temperature of a pound of water (weighed in vacuo and 
taken at between 55 and 60) by i Fahr. requires for its 
evolution the expenditure of a mechanical force [energy] 
represented by the fall of 772 Ib. through the space of one foot. 

In other words, the potential energy of 772 pounds 
elevated one foot above the ground is equivalent to 
the quantity of heat necessary to raise the temperature 
of one pound of water from 55 F. to 56 F. Later ex- 
perimenters were capable of somewhat greater accuracy, 


but the mechanical equivalent of heat is essentially what 
Joule found in his pioneer work. 

Once this important work was done, further pro- 
gress was rapid. It was soon recognized that these kinds 
of energy, mechanical and heat, are only two of its 
many forms. Everything which can be converted into 
either of them is also a form of energy. The radiation 
given off by the sun is energy, for part of it is trans- 
formed into heat on the earth. An electric current 
possesses energy, for it heats a wire or turns the wheels of 
a motor. Coal represents chemical energy, liberated as 
heat when the coal burns. In every event in nature one 
form of energy is being converted into another, always 
at some well-defined rate of exchange. In a closed 
system, one isolated from external influences, the energy 
is conserved and thus behaves like a substance. The 
sum of all possible forms of energy in such a system is 
constant, although the amount of any one kind may be 
changing. If we regard the whole universe as a closed 
system, we can proudly announce with the physicists 
of the nineteenth century that the energy of the uni- 
verse is invariant, that no part of it can ever be created 
or destroyed. 

Our two concepts of substance are, then, matter and 
energy. Both obey conservation laws: An isolated sys- 
tem cannot change either in mass or in total energy. 
Matter has weight but energy is weightless. We have 
therefore two different concepts and two conservation 
laws. Are these ideas still to be taken seriously? Or has 
this apparently well-founded picture been changed in 


the light of newer developments? It has! Further 
changes in the two concepts are connected with the 
theory of relativity. We shall return to this point later. 


The results of scientific research very often force a 
change in the philosophical view of problems which 
extend far beyond the restricted domain of science it- 
self. What is the aim of science? What is demanded of 
a theory which attempts to describe nature? These 
questions, although exceeding the bounds of physics, 
are intimately related to it, since science forms the 
material from which they arise. Philosophical gener- 
alizations must be founded on scientific results. Once 
formed and widely accepted, however, they very often 
influence the further development of scientific thought 
by indicating one of the many possible lines of pro- 
cedure. Successful revolt against the accepted view 
results in unexpected and completely different develop- 
ments, becoming a source of new philosophical aspects. 
These remarks necessarily sound vague and pointless 
until illustrated by examples quoted from the history 
of physics. 

We shall here try to describe the first philosophical 
ideas on the aim of science. These ideas greatly in- 
fluenced the development of physics until nearly a 
hundred years ago, when their discarding was forced 
by new evidence, new facts and theories, which in 
their turn formed a new background for science. 


In the whole history of science from Greek philo- 
sophy to modern physics there have been constant 
attempts to reduce the apparent complexity of natural 
phenomena to some simple fundamental ideas and re- 
lations. This is the underlying principle of all natural 
philosophy. It is expressed even in the work of the 
Atomists. Twenty-three centuries ago Democritus wrote : 

By convention sweet is sweet, by convention bitter is 
bitter, by convention hot is hot, by convention cold is cold, 
by convention colour is colour. But in reality there are 
atoms and the void. That is, the objects of sense are supposed 
to be real and it is customary to regard them as such, 
but in truth they are not. Only the atoms and the void are 

This idea remains in ancient philosophy nothing 
more than an ingenious figment of the imagination. 
Laws of nature relating subsequent events were un- 
known to the Greeks. Science connecting theory and 
experiment really began with the work of Galileo. We 
have followed the initial clues leading to the laws of 
motion. Throughout two hundred years of scientific 
research force and matter were the underlying con- 
cepts in all endeavours to understand nature. It is 
impossible to imagine one without the other because 
matter demonstrates its existence as a source of force 
by its action on other matter. 

Let us consider the simplest example: two particles 
with forces acting between them. The easiest forces to 
imagine are those of attraction and repulsion. In both 
cases the force vectors lie on a line connecting the 


material points. The demand for simplicity leads to the 
picture of particles attracting or repelling each other; 


* * 

any other assumption about the direction of the acting 
forces would give a much more complicated picture. 
Can we make an equally simple assumption about the 
length of the force vectors? Even if we want to avoid 
too special assumptions we can still say one thing: the 
force between any two given particles depends only 
on the distance between them, like gravitational forces. 
This seems simple enough. Much more complicated 
forces could be imagined, such as those which might 
depend not only on the distance but also on the velo- 
cities of the two particles. With matter and force as our 
fundamental concepts, we can hardly imagine simpler 
assumptions than that forces act along the line con- 
necting the particles and depend only on the distance. 
But is it possible to describe all physical phenomena by 
forces of this kind alone? 

The great achievements of mechanics in all its 
branches, its striking success in the development of 
astronomy, the application of its ideas to problems ap- 
parently different and non-mechanical in character, all 
these things contributed to the belief that it is possible 


to describe all natural phenomena in terms of simple 
forces between unalterable objects. Throughout the two 
centuries following Galileo's time such an endeavour, 
conscious or unconscious, is apparent in nearly all 
scientific creation. This was clearly formulated by 
Helmholtz about the middle of the nineteenth century : 

Finally, therefore, we discover the problem of physical 
material science to be to refer natural phenomena back to 
unchangeable attractive and repulsive forces whose intensity 
depends wholly upon distance. The solubility of this pro- 
blem is the condition of the complete comprehensibility of 

Thus, according to Helmholtz, the line of development 
of science is determined and follows strictly a fixed 
course : 

And its vocation will be ended as soon as the reduction 
of natural phenomena to simple forces is complete and the 
proof given that this is the only reduction of which the 
phenomena are capable. 

This view appears dull and naive to a twentieth- 
century physicist. It would frighten him to think that 
the great adventure of research could be so soon finished, 
and an unexciting if infallible picture of the universe 
established for all time. 

Although these tenets would reduce the description 
of all events to simple forces, they do leave open the 
question of just how the forces should depend on dis- 
tance. It is possible that for different phenomena this 
dependence is different. The necessity of introducing 
many different kinds of force for different events is 


certainly unsatisfactory from a philosophical point of 
view. Nevertheless this so-called mechanical view, most 
clearly formulated by Helmholtz, played an important 
role in its time. The development of the kinetic theory 
of matter is one of the greatest achievements directly 
influenced by the mechanical view. 

Before witnessing its decline, let us provisionally ac- 
cept the point of view held by the physicists of the past 
century and see what conclusions we can draw from 
their picture of the external world. 

Is it possible to explain the phenomena of heat in 
terms of the motions of particles interacting through 
simple forces? A closed vessel contains a certain mass 
of gas air, for example at a certain temperature. By 
heating we raise the temperature, and thus increase the 
energy. But how is this heat connected with motion? 
The possibility of such a connection is suggested both 
by our tentatively accepted philosophical point of 
view and by the way in which heat is generated by 
motion. Heat must be mechanical energy if every 
problem is a mechanical one. The gljr/it flf *^r kmtic 
theory is to present the cop^p^ of matVr just in this 
way. According to this theory a gas is a congregation 

Of ar> 

moving in all directions T colliding with^ each other and 
changing in direction of motiorpwith each collision. 
There must exist an average speed of molecules, just as 
in a large human community there exists an average 


age, or an average wealth. There will therefore^ be an 
average kinetic energy per particle. More heat in the 
vessel means a greater ^verageTffietic energy. Thus 
heat, accofdlilgTo tKiiTpicture, is not a special form of 
energy different from the mechanical one but is just 
the kinetic energy of molecular motion. To any definite 
temperature there corresponds a definite average kinetic 
energy per molecule. This is, in fact, not an arbitrary 
assumption. We are forced to regard the kinetic energy 
of a molecule as a measure of the temperature of the 
gas if we wish to form a consistent mechanical picture 
of matter. 

This theory is more than a play of the imagination. 
It can be shown that the kinetic theory of gases is not 
only in agreement with experiment, but actually leads 
to a more profound understanding of the facts. This 
may be illustrated by a few examples. 

We have a vessel closed by a piston which can move 
freely. The vessel contains a certain amount of gas to 
be kept at a constant temperature. If the piston is ini- 
tially at rest in some position, it can be moved upward 
by removing and downward by adding weight. To 
push the piston down force must be used acting against 
the inner pressure of the gas. What is the mechanism 
of this inner pressure according to the kinetic theory? 
A tremendous number of particles constituting the gas 
are moving in all directions. They bombard the walls 
and the piston, bouncing back like balls thrown against 
a wall. This continual bombardment by a great number 
of particles keeps the piston at a certain height by 


opposing the force of gravity acting downward on 
the piston and the weights. In one direction there is a 
constant gravitational force, in the other very many 
irregular blows from the molecules. The net effect on 
the piston of all these small irregular forces must be 
equal to that of the force of gravity if there is to be 



Suppose the piston were pushed down so as to com- 
press the gas to a fraction of its former volume, say 
one-half, its temperature being kept unchanged. What, 
according to the kinetic theory, can we expect to hap- 
pen? Will the force due to the bombardment be more 
or less effective than before? The particles are now 
packed more closely. Although the average kinetic 
energy is still the same, the collisions of the particles 
with the piston will now occur more frequently and 
thus the total force will be greater. It is clear from this 


picture presented by the kinetic theory that to keep 
the piston in this lower position more weight is re- 
quired. This simple experimental fact is well known, 
but its prediction follows logically from the kinetic 
view of matter. 

Consider another experimental arrangement. Take 
two vessels containing equal volumes of different gases, 
say hydrogen and nitrogen, both at the same tempera- 
ture. Assume the two vessels are closed with identical 
pistons, on which are equal weights. This means, briefly, 
that both gases have the same volume, temperature, 
and pressure. Since the temperatue is the same, so, 
according to the theory, is the average kinetic energy 
per particle. Since the pressures are equal, the two 
pistons are bombarded with the same total force. On 
the average every particle carries the same energy and 
both vessels have the same volume. Therefore, the 
number of molecules in each must be the same, although the 
gases are chemically different. This result is very im- 
portant for the understanding of many chemical pheno- 
mena. It means that the number of molecules in a 
given volume, at a certain temperature and pressure, 
is something which is characteristic, not of a particular 
gas, but of all gases. It is most astonishing that the 
kinetic theory not only predicts the existence of such a 
universal number, but enables us to determine it. To 
this point we shall return very soon. 

The kinetic theory of matter explains quantitatively 
as well as qualitatively the laws of gases as determined 
by experiment. Furthermore, the theory is not restricted 


to gases, although its greatest successes have been in 
this domain. 

A gas can be liquefied by means of a decrease of tem- 
perature. A fall in the temperature of matter means a 
decrease in the average kinetic energy of its particles. 
It is therefore clear that the average kinetic energy of 
a liquid particle is smaller than that of a corresponding 
gas particle. 

A striking manifestation of the motion of particles 
in liquids was given for the first time by the so-called 
Brownian movement, a remarkable phenomenon which 
would remain quite mysterious and incomprehensible 
without the kinetic theory of matter. It was first ob- 
served by the botanist Brown, and was explained eighty 
years later, at the beginning of this century. The only 
apparatus necessary for observing Brownian motion is 
a microscope, which need not even be a particularly 
good one. 

Brown was working with grains of pollen of certain 
plants, that is: 

particles or granules of unusually large size varying from 
one four-thousandth to about five-thousandth of an inch in 

He reports further : 

While examining the form of these particles immersed in 
water, I observed many of them evidently in motion .... 
These motions were such as to satisfy me, after frequently 
repeated observation, that they arose neither from current 
in the fluid nor from its gradual evaporation, but belonged 
to the particle itself. 


What Brown observed was the unceasing agitation 
of the granules when suspended in water and visible 
through the microscope. It is an impressive sight ! 

Is the choice of particular plants essential for the 
phenomenon? Brown answered this question by re- 
peating the experiment with many different plants, 
and found that all the granules, if sufficiently small, 
showed such motion when suspended in water. Further- 
more, he found the same kind of restless, irregular 
motion in very small particles of inorganic as well as 
organic substances. Even with a pulverized fragment 
of a sphinx he observed the same phenomenon ! 

How is this motion to be explained? It seems con- 
tradictory to all previous experience. Examination of 
the position of one suspended particle, say every thirty 
seconds, reveals the fantastic form of its path. The 
amazing thing is the apparently eternal character of 
the motion. A swinging pendulum placed in water 
soon comes to rest if not impelled by some external 
force. The existence of a never-diminishing motion 
seems contrary to all experience. This difficulty was 
splendidly clarified by the kinetic theory of matter. 

Looking at water through even our most powerful 
microscopes we cannot see molecules and their motion 
as pictured by the kinetic theory of matter. It must be 
concluded that if the theory of water as a congrega- 
tion of particles is correct, the size of the particles must 
be beyond the limit of visibility of the best micro- 
scopes. Let us nevertheless stick to the theory and as- 
sume that it represents a consistent picture of reality. 


The Brownian particles visible through a microscope 
are bombarded by the smaller ones composing the 
water itself. The Brownian movement exists if the 
bombarded particles are sufficiently small. It exists 
because this bombardment is not uniform from all sides 
and cannot be averaged out, owing to its irregular and 
haphazard character. The observed motion is thus the 
result of the unobservable one. The behaviour of the 
big particles reflects in some way that of the molecules, 
constituting, so to speak, a magnification so high that 
it becomes visible through the microscope. The irregular 
and haphazard character of the path of the Brownian 
particles reflects a similar irregularity in the path of 
the smaller particles which constitute matter. We can 
understand, therefore, that a quantitative study of 
Brownian movement can give us deeper insight into 
the kinetic theory of matter. It is apparent that the 
visible Brownian motion depends on the size of the 
invisible bombarding molecules. There would be no 
Brdwnian motion at all if the bombarding molecules 
did not possess a certain amount of energy or, in other 
words, if they did not have mass and velocity. That 
the study of Brownian motion can lead to a deter- 
mination of the mass of a molecule is therefore not 

Through laborious research, both theoretical and 
experimental, the quantitative features of the kinetic 
theory were formed. The clue originating in the phe- 
nomenon of Brownian movement was one of those 
which led to the quantitative data. The same data can 

EE 3 


be obtained in different ways, starting from quite dif- 
ferent clues. The fact that all these methods support 
the same view is most important, for it demonstrates 
the internal consistency of the kinetic theory of matter. 

Only one of the many quantitative results reached 
by experiment and theory will be mentioned here. 
Suppose we have a gram of the lightest of all elements, 
hydrogen, and ask: how many particles are there in 
this one gram? The answer will characterize not only 
hydrogen but also all other gases, for we already know 
under what conditions two gases have the same number 
of particles. 

The theory enables us to answer this question from 
certain measurements of the Brownian motion of a 
suspended particle. The answer is an astonishingly 
great number: a three followed by twenty- three other 
digits ! The number of molecules in one gram of hydro- 
gen is 303,000,000,000,000,000,000,000. 

Imagine the molecules of a gram of hydrogen so in- 
creased in size that they are visible through a microscope : 
say that the diameter becomes one five-thousandth 
of an inch, such as that of a Brownian particle. Then, 
to pack them closely, we should have to use a box 
each side of which is about one-quarter of a mile long ! 

We can easily calculate the mass of one such hydrogen 
molecule by dividing i by the number quoted above. 
The answer is a fantastically small number : 

o-ooo ooo ooo ooo ooo ooo ooo 0033 gram, 
representing the mass of one molecule of hydrogen. 


Brownian particles seen through a microscope 

One Brownian particle photo- 
graphed by a long exposure and 
covering a surface 

Consecutive positions 
observed for one of the 
Brownian particles 

The path avenged 
from these conse- 
cutive positions 


The experiments on Brownian motion are only some 
of the many independent experiments leading to the 
determination of this number which plays such an im- 
portant part in physics. 

In the kinetic theory of matter and in all its impor- 
tant achievements we see the realization of the general 
philosophical programme: to reduce the explanation of 
all phenomena to the interaction between particles of 


In mechanics the future path of a moving body can be 
predicted and its past disclosed if its present condition and 
the forces acting upon it are known. Thus, for example, the 
future paths of all planets can be foreseen. The active forces 
are Newton's gravitational forces depending on the distance 
alone. The great results of classical mechanics suggest that 
the mechanical view can be consistently applied to all branches 
of physics, that all phenomena can be explained by the action 
of forces representing either attraction or repulsion, depending 
only upon distance and acting between unchangeable par- 

In the kinetic theory of matter we see how this view, arising 
from mechanical problems, embraces the phenomena of heat 
and how it leads to a successful picture of the structure of 



The two electric fluids The magnetic fluids The first serious 
difficulty The velocity of light Light as a substance The 
riddle of colour What is a wave? The wave theory of 
light Longitudinal or transverse light waves? Ether and 
the mechanical view 


THE following pages contain a dull report of some 
very simple experiments. The account will be boring 
not only because the description of experiments is un- 
interesting in comparison with their actual performance, 
but also because the meaning of the experiments does 
not become apparent until theory makes it so. Our 
purpose is to furnish a striking example of the role of 
theory in physics. 

i . A metal bar is supported on a glass base, and each 
end of the bar is connected by means of a wire to an 
electroscope. What is an electroscope? It is a simple 
apparatus consisting essentially of two leaves of gold 
foil hanging from the end of a short piece of metal. 
This ii enclosed in a glass jar or flask and the metal is 
in contact only with non-metallic bodies, called in- 
sulators. In addition to the electroscope and metal bar 
we are equipped with a hard rubber rod and a piece of 

The experiment is performed as follows: we look 


to see whether the leaves hang close together, for this 
is their normal position. If by chance they do not, a 
touch of the finger on the metal rod will bring them 
together. These preliminary steps being taken, the 

rubber rod is rubbed vigorously with the flannel and 
brought into contact with the metal. The leaves sepa- 
rate at once ! They remain apart even after the rod is 

2. We perform another experiment, using the same 
apparatus as before, again starting with the gold leaves 
hanging close together. This time we do not bring the 
rubbed rod into actual contact with the metal, but 
only near it. Again the leaves separate. But there is a 
difference ! When the rod is taken away without having 
touched the metal, the leaves immediately fall back to 
their normal position instead of remaining separated. 

3. Let us change the apparatus slightly for a third 
experiment. Suppose that the metal bar consists of two 
pieces, joined together. We rub the rubber rod with 


flannel and again bring it near the metal. The same 
phenomenon occurs, the leaves separate. But now let 

us divide the metal rod into its two separate parts and 
then take away the rubber rod. We notice that in this 
case the leaves remain apart, instead of falling back to 
their normal position as in the second experiment. 

It is difficult to evince enthusiastic interest in these 
simple and naive experiments. In the Middle Ages their 
performer would probably have been condemned; to 
us they seem both dull and illogical. It would be very 
difficult to repeat them, after reading the account only 
once, without becoming confused. Some notion of the 
theory makes them understandable. We could say more : 
it is hardly possible to imagine such experiments per- 
formed as accidental play, without the pre-existence of 
more or less definite ideas about their meaning. 

We shall now point out the underlying ideas of a very 
simple and naive theory which explains all the facts 

There exist two electric fluids, one called positive ( 4- ) 
and the other negative ( ). They are somewhat like 
substance in the sense already explained, in that the 
amount can be enlarged or diminished, but the total 
in any isolated system is preserved. There is, how- 


ever, an essential difference between this case and that 
of heat, matter or energy. We have two electrical 
substances. It is impossible here to use the previous 
analogy of money unless it is somehow generalized. A 
body is electrically neutral if the positive and negative 
electric fluids exactly cancel each other. A man has 
nothing, either because he really has nothing, or because 
the amount of money put aside in his safe is exactly 
equal to the sum of his debts. We can compare the debit 
and credit entries in his ledger to the two kinds of 
electric fluids. 

The next assumption of the theory is that two electric 
fluids of the same kind repel each other, while two of 
the opposite kind attract. This can be represented 
graphically in the following way : 


A final theoretical assumption is necessary: There 
are two kinds of bodies, those in which the fluids can 
move freely, called conductors, and those in which they 
cannot, called insulators. As is always true in such cases, 
this division is not to be taken too seriously. The 
ideal conductor or insulator is a fiction which can 


never be realized. Metals, the earth, the human body, 
are all examples of conductors, although not equally 
good. Glass, rubber, china, and the like are insulators. 
Air is only partially an insulator, as everyone who has 
seen the described experiments kqows. It is always a 
good excuse to ascribe the bad results of electrostatic 
experiments to the humidity of the air, which increases 
its conductivity. 

These theoretical assumptions are sufficient to explain 
the three experiments described. We shall discuss them 
once more, in the same order as before, but in the light 
of the theory of electric fluids. 

i . The rubber rod, like all other bodies under normal 
conditions, is electrically neutral. It contains the two 
fluids, positive and negative, in equal amounts. By 
rubbing with flannel we separate them. This statement 
is pure convention, for it is the application of the ter- 
minology created by the theory to the description of 
the process of rubbing. The kind of electricity that the 
rod has in excess afterwards is called negative, a name 
which is certainly only a matter of convention. If the 
experiments had been performed with a glass rod 
rubbed with cat's fur we should have had to call the 
excess positive, to conform with the accepted conven- 
tion. To proceed with the experiment, we bring electric 
fluid to the metal conductor by touching it with the 
rubber. Here it moves freely, spreading over the whole 
metal including the gold leaves. Since the action of 
negative on negative is repulsion, the two leaves try to 
get as far from each other as possible and the result is 


the observed separation. The metal rests on glass or 
some other insulator so that the fluid remains on the 
conductor, as long as the conductivity of the air permits. 
We understand now why we have to touch the metal 
before beginning the experiment. In this case the metal, 
the human body, and the earth form one vast conductor, 
with the electric fluid so diluted that practically nothing 
remains on the electroscope. 

2. This experiment begins just in the same way as 
the previous one. But instead of being allowed to touch 
the metal the rubber is now only brought near it. The 
two fluids in the conductor, being free to move, are 
separated, one attracted and the other repelled. They 
mix again when the rubber rod is removed, as fluids of 
opposite kinds attract each other. 

3. Now we separate the metal into two parts and 
afterwards remove the rod. In this case the two fluids 
cannot mix, so that the gold leaves retain an excess of 
one electric fluid and remain apart. 

In the light of this simple theory all the facts men- 
tioned here seem comprehensible. The same theory does 
more, enabling us to understand not only these, but 
many other facts in the realm of "electrostatics". The 
aim of every theory is to guide us to new facts, sug- 
gest new experiments, and lead to the discovery of new 
phenomena and new laws. An example will make this 
:lear. Imagine a change in the second experiment. 
Suppose I keep the rubber rod near the metal and at 
the same time touch the conductor with my finger. 
What will happen now? Theory answers: the repelled 


fluid ( ) can now make its escape through my body, 
with the result that only one fluid remains, the positive. 

Only the leaves of the electroscope near the rubber rod 
will remain apart. An actual experiment confirms this 

The theory with which we are dealing is certainly 
naive and inadequate from the point of view of modern 
physics. Nevertheless it is a good example showing the 
characteristic features of every physical theory. 

There are no eternal theories in science. It always 
happens that some of the facts predicted by a theory 
are disproved by experiment. Every theory has its 
period of gradual development and triumph, after 
which it may experience a rapid decline. The rise and 
fall of the substance theory of heat, already discussed 
here, is one of many possible examples. Others, more 
profound and important, will be discussed later. Nearly 
every great advance in science arises from a crisis in the 
old theory, through an endeavour to find a way out of 
the difficulties created. We must examine old ideas, old 


theories, although they belong to the past, for this is the 
only way to understand the importance of the new ones 
and the extent of their validity. 

In the first pages of our book we compared the role 
of an investigator to that of a detective who, after 
gathering the requisite facts, finds the right solution 
by pure thinking. In one essential this comparison must 
be regarded as highly superficial. Both in life and in 
detective novels the crime is given. The detective must 
look for letters, fingerprints, bullets, guns, but at least 
he knows that a murder has been committed. This is 
not so for a scientist. It should not be difficult to 
imagine someone who knows absolutely nothing about 
electricity, since all the ancients lived happily enough 
without any knowledge of it. Let this man be given 
metal, gold foil, bottles, hard-rubber rod, flannel, in 
short, all the material required for performing our 
three experiments. He may be a very cultured person, 
but he will probably put wine into the bottles, use the 
flannel for cleaning, and never once entertain the idea 
of doing the things we have described. For the detec- 
tive the crime is given, the problem formulated: who 
killed Cock Robin? The scientist must, at least in part, 
commit his own crime, as well as carry out the investi- 
gation. Moreover, his task is not to explain just one 
case, but all phenomena which have happened or may 
still happen. 

In the introduction of the concept of fluids we see 
the influence of those mechanical ideas which attempt 
to explain everything by substances and simple forces 


acting between them. To see whether the mechanical 
point of view can be applied to the description of 
electrical phenomena, we must consider the following 
problem. Two small spheres are given, both with an 
electric charge, that is, both carrying an excess of one 
electric fluid. We know that the spheres will either 
attract or repel each other. But does the force depend 
only on the distance, and if so, how? The simplest 
guess seems to be that this force depends on the dis- 
tance in the same way as gravitational force, which 
diminishes, say, to one-ninth of its former strength if the 
distance is made three times as great. The experiments 
performed by Coulomb showed that this law is really 
valid. A hundred years after Newton discovered the 
law of gravitation, Coulomb found a similar dependence 
of electrical force on distance. The two major differences 
between Newton's law and Coulomb's law are : gravita- 
tional attraction is always present, while electric forces 
exist only if the bodies possess electric charges. In the 
gravitational case there is only attraction, but electric 
forces may either attract or repel. 

There arises here the same question which we con- 
sidered in connection with heat. Are the electrical 
fluids weightless substances or not? In other words, is 
the weight of a piece of metal the same whether neutral 
or charged? Our scales show no difference, ^e con- 
clude that the electric fluids are also members of the 
family of weightless substances^ 

Further progress in the theory of electricity requires 
the introduction of two new concepts. Again we shall 


avoid rigorous definitions, using instead analogies with 
concepts already familiar. We remember how essential 
it was for an understanding of the phenomena of heat 
to distinguish between heat itself and temperature. It 
is equally important here to distinguish between electric 
potential and electric charge. The difference between 
the two concepts is made clear by the analogy : 

7 Electric potential Temperature 
Electric charge Heat / 

Two conductors, for example two spheres of dif- 
ferent size, may have the same electric charge, that is 
the same excess of one electric fluid, but the potential 
will be different in the two cases, being higher for the 
smaller and lower for the larger sphere. The electric 
fluid will have greater density and^thus be more com- 
pressed on the small conductor. Since the repulsive 
forces must increase with the density, the tendency of 
the charge to escape will be greater in the case of the 
smaller sphere than in that of the larger. This tendency 
of charge to escape from a conductor is a direct measure 
of its potentials/In order to show clearly the difference 
between charge and potential we shall formulate a few 
sentences describing the behaviour of heated bodies, 
and the corresponding sentences concerning charged 


Two bodies, initially at dif- Two insulated conductors, 
ferent temperatures, reach initially at different electric 


the same temperature after potentials, very quickly reach 
some time if brought into the same potential if brought 
contact. into contact. 

Equal quantities of heat Equal amounts of electric 

produce different changes of charge produce differem 

temperature in two bodies changes of electric potentia 

if their heat capacities are in two bodies if their elec 

different. trical capacities are different 

A thermometer in contact An electroscope in contact 

with a body indicates by with a conductor indicates 

the length of its mercury by the separation of the 

column its own tempera- gold leaves its own elec- 

ture and therefore the tern- trie potential and therefore 

perature of the body. the electric potential of the 


But this analogy must not be pushed too far. An 
example shows the differences as well as the similarities. 
If a hot body is brought into contact with a cold one, 
the heat flows from the hotter to the colder. On the 
other hand, suppose that we have two insulated con- 
ductors having equal but opposite charges, one positive 
and the other negative. The two are at different 
potentials. By convention we regard the potential 
corresponding to a negative charge as lower than that 
corresponding to a positive charge. If the two con- 
ductors are brought together or connected by a wire, it 
follows from the theory of electric fluids that they will 
show no charge and thus no difference of electric poten- 
tial at all. We must imagine a "flow" of electric charge 
from one conductor to the other during the short time 
in which the potential difference is equalized. But how? 


Does the positive fluid flow to the negative body, or the 
negative fluid to the positive body? 

In the material presented here we have no basis for 
deciding between these two alternatives. We can as- 
sume either of the two possibilities, or that the flow is 
simultaneous in both directions. It is only a matter of 
adopting a convention, and no significance can be 
attached to the choice, for we know no method of 
deciding the question experimentally. Further develop- 
ment leading to a much more profound theory of 
electricity gave an answer to this problem, which is 
quite meaningless when formulated in terms of the 
simple and primitive theory of electric fluids. Here we 
shall simply adopt the following mode of expression. 
The electric fluid flows from the conductor having the 
higher potential to that having the lower. In the case 
of our two conductors, the electricity thus flows from 

positive to negative. This expression is only a matter of 
convention and is at this point quite arbitrary. The 
whole difficulty indicates that the analogy between heat 
and electricity is by no means complete. 

We have seen the possibility of adapting the me- 
chanical view to a description of the elementary facts 
of electrostatics. The same is possible in the case of 
magnetic phenomena. 



We shall proceed here in the same manner as before, 
starting with very simple facts and then seeking their 
theoretical explanation. 

i. We have two long bar magnets, one suspended 
freely at its centre, the other held in the hand. The 
ends of the two magnets are brought together in such 
a way that a strong attraction is noticed between them. 
This can always be done. If no attraction results, we 
must turn the magnet and try the other end. Some- 
thing will happen if the bars are magnetized at all. The 

ends of the magnets are called their poles. To continue 
with the experiment we move the pole of the magnet 
held in the hand along the other magnet. A decrease 
in the attraction is noticed and when the pole reaches 
the middle of the suspended magnet there is no evi- 
dence of any force at all. If the pole is moved farther 
in the same direction a repulsion is observed, attaining 
its greatest strength at the second pole of the hanging 

2. The above experiment suggests another. Each 


magnet has two poles. Can we not isolate one of them? 
The idea is very simple : just break a magnet into two 
equal parts. We have seen that there is no force between 
the pole of one magnet and the middle of the other. 
But the result of actually breaking a magnet is sur- 
prising and unexpected. If we repeat the experiment 
described under i , with only half a magnet suspended, 
the results are exactly the same as before ! Where there 
was no trace of magnetic force previously, there is now 
a strong pole. 

How are these facts to be explained? We can attempt 
to pattern a theory of magnetism after the theory of 
electric fluids. This is suggested by the fact that here, 
as in electrostatic phenomena, we have attraction and 
repulsion. Imagine two spherical conductors possessing 
equal charges, one positive and the other negative. 
Here "equal" means having the same absolute value; 
+ 5 and 5, for example, have the same absolute 
value. Let us assume that these spheres are connected 

by means of an insulator such as a glass rod. Schemati- 
cally this arrangement can be represented by an arrow 
directed from the negatively charged conductor to the 
positive one. We shall call the whole thing an electric 
dipole. It is clear that two such dipoles would behave 
exactly like the bar magnets in experiment i . If we 
think of our invention as a model for a real magnet, 


we may say, assuming the existence of magnetic fluids, 
that a magnet is nothing but a magnet dipole y having 
at its ends two fluids of different kinds. This simple 
theory, imitating the theory of electricity, is adequate 
for an explanation of the first experiment. There would 
be attraction at one end, repulsion at the other, and 
a balancing of equal and opposite forces in the middle. 
But what of the second experiment? By breaking 
the glass rod in the case of the electric dipole we get 
two isolated poles. The same ought to hold good for 
the iron bar of the magnetic dipole, contrary to the 
results of the second experiment. Thus this contra- 
diction forces us to introduce a somewhat more subtle 
theory. Instead of our previous model we may imagine 
that the magnet consists of very small elementary mag- 
netic dipoles which cannot be broken into separate 
poles. Order reigns in the magnet as a whole, for all 
the elementary dipoles are directed in the same way. 

We see immediately why cutting a magnet causes two 
new poles to appear on the new ends, and why this 
more refined theory explains the facts of experiment i 
as well as 2. 

For many facts, the simpler theory gives an explana- 
tion and the refinement seems unnecessary. Let us take 
an example : We know that a magnet attracts pieces of 


iron. Whyr In a piece of ordinary iron the two mag- 
netic fluids are mixed, so that no net effect results. 
Bringing a positive pole near acts as a "command of 
division" to the fluids, attracting the negative fluid of 
the iron and repelling the positive. The attraction 
between iron and magnet follows^If the magnet is 
removed, the fluids go back to more or less their original 
state, depending on the extent to which they remember 
the commanding voice of the external force. 

Little need be said about the quantitative side of the 
problem. With two very long magnetized rods we could 
investigate the attraction (or repulsion) of their poles 
when brought near one another. The effect of the other 
ends of the rods is negligible if the rods are long enough. 
How does the attraction or repulsion depend on the 
distance between the poles? The answer given by 
Coulomb's experiment is that this dependence on dis- 
tance is the same as in Newton's law of gravitation and 
Coulomb's law of electrostatics. 

We see again in this theory the application of a 
general point of view: the tendency to describe all 
phenomena by means of attractive and repulsive forces 
depending only on distance and acting betweenr un- 
changeable particles. 

One well-known fact should be mentioned, for later 
we shall make use of it. The earth is a great magnetic 
dipole. There is not the slightest trace of an explanation 
as to why this is true. The North Pole is approximately 
the minus ( ) and the South Pole the plus ( -f ) mag- 
netic pole of the earth. The names plus and minus are 


only a matter of convention, but when once fixed, 
enable us to designate poles in any other case. A magnetic 
needle supported on a vertical axis obeys the command 
of the magnetic force of the earth. It directs its ( + ) 
pole toward the North Pole, that is, toward the (-) 
magnetic pole of the earth. 

Although we can consistently carry out the mechanical 
view in the domain of electric and magnetic phenomena 
introduced here, there is no reason to be particularly 
proud or pleased about it. Some features of the theory 
are certainly unsatisfactory if not discouraging. New 
kinds of substances had to be invented: two electric 
fluids and the elementary magnetic dipoles. The wealth 
of substances begins to be overwhelming ! 

The forces are simple. They are expressible in a 
similar way for gravitational, electric, and magnetic 
forces. But the price paid for this simplicity is high: 
the introduction of new weightless substances. These 
are rather artificial concepts, and quite unrelated to the 
fundamental substance, mass. 


We are now ready to note the first grave difficulty in 
the application of our general philosophical point of 
view. It will be shown later that this difficulty, together 
with another even more serious, caused a complete 
breakdown of the belief that all phenomena can be 
explained mechanically. 

The tremendous development of electricity as a 
branch of science and technique began with the dis- 


covery of the electric current. Here we find in the 
history of science one of the very few instances in which 
accident seemed to play an essential role. The story of 
the convulsion of a frog's leg is told in many different 
ways. Regardless of the truth concerning details, there 
is no doubt that Galvani's accidental discovery led Volt a 
at the end of the eighteenth century to the construction 
of what is known as a voltaic battery. This is no longer of 
any practical use, but it still furnishes a very simple 
example of a source of current in school demonstrations 
and in textbook descriptions. 

The principle of its construction is simple. There are 
several glass tumblers, each containing water with a 
little sulphuric acid. In each glass two metal plates, one 
copper and the other zinc, are immersed in the solution. 
The copper plate of one glass is connected to the zinc of 
the next, so that only the zinc plate of the first and the 
copper plate of the last glass remain unconnected. We 
can detect a difference in electric potential between the 
copper in the first glass and the zinc in the last by means 
of a fairly sensitive electroscope if the number of the 
"elements 55 , that is, glasses with plates, constituting the 
battery, is sufficiently large. 

It was only for the purpose of obtaining something 
easily measurable with apparatus already described that 
we introduced a battery consisting of several elements. 
For further discussion, a single element will serve just 
as well. The potential of the copper turns out to be 
higher than that of the zinc. "Higher" is used here 
in the sense in which +2 is greater than -2. If one 


conductor is connected to the free copper plate and 
another to the zinc, both will become charged, the first 
positively and the other negatively. Up to this point 
nothing particularly new or striking has appeared, and 
we may try to apply our previous ideas about potential 
differences. We have seen that a potential difference 
between two conductors can be quickly nullified by 
connecting them with a wire, so that there is a flow 
of electric fluid from one conductor to the other. This 
process was similar to the equalization of temperatures 
by heat flow. But does this work in the case of a voltaic 
battery? Volta wrote in his report that the plates behave 
like conductors : 

. . . feebly charged, which act unceasingly or so that their 
charge after each discharge re-establishes itself; which, in a 
word, provides an unlimited charge or imposes a perpetual 
action or impulsion of the electric fluid. 

The astonishing result of his experiment is that the 
potential difference between the copper and zinc plates 
does not vanish as in the case of two charged conduc- 
tors connected by a wire. The difference persists, and 
according to the fluids theory it must cause a constant 
flow of electric fluid from the higher potential level 
(copper plate) to the lower (zinc plate). In an attempt 
to save the fluid theory, we may assume that some con- 
stant force acts to regenerate the potential difference 
and cause a flow of electric fluid. But the whole pheno- 
menon is astonishing from the standpoint of energy. 
A noticeable quantity of heat is generated in the wire 
carrying the current, even enough to melt the wire if 


it is a thin one. Therefore, heat-energy is created in the 
wire. But the whole voltaic battery forms an isolated 
system, since no external energy is being supplied. If 
we want to save the law of conservation of energy we 
must find where the transformations take place, and at 
what expense the heat is created. It is not difficult to 
realize that complicated chemical processes are taking 
place in the battery, processes in which the immersed 
copper and zinc, as well as the liquid itself, take active 
parts. From the standpoint of energy this is the chain 
of transformations which are taking place: chemical 
energy -> energy of the flowing electric fluid, i.e., the 
current -> heat. A voltaic battery does not last for ever; 
the chemical changes associated with the flow of elec- 
tricity make the battery useless after a time. 

The experiment which actually revealed the great 
difficulties in applying the mechanical ideas must sound 
strange to anyone hearing about it for the first time. It 
was performed by Oersted about a hundred and twenty 
years ago. He reports : 

By these experiments it seems to be shown that the mag- 
netic needle was moved from its position by help of a 
galvanic apparatus, and that, when the galvanic circuit was 
closed, but not when open, as certain very celebrated 
physicists in vain attempted several years ago. 

Suppose we have a voltaic battery and a conducting 
wire. If the wire is connected to the copper plate but 
not to the zinc, there will exist a potential difference 
but no current can flow. Let us assume that the wire is 
bent to form a circle, in the centre of which a magnetic 


needle is placed, both wire and needle lying in the 
same plane. Nothing happens so long as the wire does 
not touch the zinc plate. There are no forces acting, 
the existing potential difference having no influence 

whatever on the position of the needle. It seems 
difficult to understand why the "very celebrated 
physicists", as Oersted called them, expected such an 

But now let us join the wire to the zinc plate. Im- 
mediately a strange thing happens. The magnetic needle 
turns from its previous position. One of its poles now 
points to the reader if the page of this book represents 
the plane of the circle. The effect is that of a force, 
perpendicular to the plane, acting on the magnetic pole. 
Faced with the facts of the experiment, we can hardly 
avoid drawing such a conclusion about the direction of 
the force acting. 

This experiment is interesting, in the first place, be- 


cause it shows a relation between two apparently quite 
different phenomena, magnetism and electric current. 
There is another aspect even more important. The force 
between the magnetic pole and the small portions of the 
wire through which the current flows cannot lie along 
lines connecting the wire and needle, or the particles 
of flowing electric fluid and the elementary magnetic 
dipoles. The force is perpendicular to these lines ! For 
the first time there appears a force quite different from 
that to which, according to our mechanical point of 
view, we intended to reduce all actions in the external 
world. We remember that the forces of gravitation, 
electrostatics, and magnetism, obeying the laws of 
Newton and Coulomb, act along the line adjoining the 
two attracting or repelling bodies. 

This difficulty was stressed even more by an experi- 
ment performed with great skill by Rowland nearly 
sixty years ago. Leaving out technical details, this 
experiment could be described as follows. Imagine a 
small charged sphere. Imagine further that this sphere 
moves very fast in a circle at the centre of which is a 
magnetic needle. This is, in principle, the same experi- 
ment as Oersted's, the only difference being that instead 
of an ordinary current we have a mechanically effected 
motion of the electric charge. Rowland found that the 
result is indeed similar to that observed when a current 
flows in a circular wire. The magnet is deflected by a 
perpendicular force. 

Let us now move the charge faster. The force acting 
on the magnetic pole is, as a result, increased; the 


deflection from its initial position becomes more distinct. 
This observation presents another grave complication. 


V \ 


Not only does the force fail to lie on the line connecting 
charge and magnet, but the intensity of the force 
depends on the velocity of the charge. The whole 
mechanical point of view was based on the belief that 
all phenomena can be explained in terms of forces 
depending only on the distance and not on the velocity. 
The result of Rowland's experiment certainly shakes 
this belief. Yet we may choose to be conservative and 
seek a solution within the frame of old ideas. 

Difficulties of this kind, sudden and unexpected ob- 
stacles in the triumphant development of a theory, arise 
frequently in science. Sometimes a simple generaliza- 
tion of the old ideas seems, at least temporarily, to be a 
good way out. It would seem sufficient in the present 
case, for example, to broaden the previous point of view 
and introduce more general forces between the elemen- 
tary particles. Very often, however, it is impossible to 
patch up an old theory, and the difficulties result in its 


downfall and the rise of a new one. Here it was not 
only the behaviour of a tiny magnetic needle which 
broke the apparently well-founded and successful me- 
chanical theories. Another attack, even more vigorous, 
came from an entirely different angle. But this is 
another story, and we shall tell it later. 


In Galileo's Two New Sciences we listen to a conversation 
of the master and his pupils about the velocity of 

SAGREDO: But of what kind and how great must we 
consider this speed of light to be? Is it instantaneous or 
momentary or does it, like other motion, require time? Can 
we not decide this by experiment? 

SIMPLICIO: Everyday experience shows that the propa- 
gation of light is instantaneous ; for when we see a piece of 
artillery fired, at great distance, the flash reaches our eyes 
without lapse of time ; but the sound reaches the ear only 
after a noticeable interval. 

SAGREDO: Well, Simplicio, the only thing I am able to 
infer from this familiar bit of experience is that sound, in 
reaching our ears, travels more slowly than light; it does not 
inform me whether the coming of the light is instantaneous 
or whether, although extremely rapid, it still occupies 
time .... 

SALVIATI: The small conclusiveness of these and other 
similar observations once led me to devise a method by 
which one might accurately ascertain whether illumination, 
i.e., propagation of light, is really instantaneous. . .. 

Salviati goes on to explain the method of his experi- 
ment. In order to understand his idea let us imagine 
that the velocity of light is not only finite, but also 


small, that the motion of light is slowed down, like that 
in a slow-motion film. Two men, A and J5, have covered 
lanterns and stand, say, at a distance of one mile from 
each other. The first man, A, opens his lantern. The 
two have made an agreement that B will open his the 
moment he sees A's light. Let us assume that in our 
"slow motion" the light travels one mile in a second. 
A sends a signal by uncovering his lantern. B sees it 
after one second and sends an answering signal. This is 
received by A two seconds after he had sent his own. 
That is to say, if light travels with a speed of one mile 
per second, then two seconds will elapse between .4's 
sending and receiving a signal, assuming that B is a 
mile away. Conversely, if A does not know the velocity 
of light but assumes that his companion kept the agree- 
ment, and he notices the opening of 5's lantern two 
seconds after he opened his, he can conclude that the 
speed of light is one mile per second. 

With the experimental technique available at that 
time Galileo had little chance of determining the 
velocity of light in this way. If the distance were a mile, 
he would have had to detect time intervals of the order 
of one hundred-thousandth of a second ! 

Galileo formulated the problem of determining the 
velocity of light, but did not solve it. The formulation 
of a problem is often more essential than its solution, 
which may be merely a matter of mathematical or 
experimental skill. To raise new questions, new possibili- 
ties, to regard old problems from a new angle, requires 
creative imagination and marks real advance in science. 


The principle of inertia, the law of conservation of 
energy were gained only by new and original thoughts 
about already well-known experiments and phenomena. 
Many instances of this kind will be found in the following 
pages of this book, where the importance of seeing known 
facts in a new light will be stressed and new theories 

Returning to the comparatively simple question of 
determining the velocity of light, we may remark that 
it is surprising that Galileo did not realize that his 
experiment could be performed much more simply and 
accurately by one man. Instead of stationing his com- 
panion at a distance he could have mounted there a 
mirror, which would automatically send back the signal 
immediately after receiving it. 

About two hundred and fifty years later this very 
principle was used by Fizeau, who was the first to deter- 
mine the velocity of light by terrestrial experiments. It 
had been determined by Roemer much earlier, though 
less accurately, by astronomical observation. 

It is quite clear that in view of its enormous mag- 
nitude, the velocity of light could be measured only 
by taking distances comparable to that between the 
earth and another planet of the solar system or by a 
great refinement of experimental technique. The first 
method was that of Roemer, the second that of Fizeau. 
Since the days of these first experiments the very im- 
portant number representing the velocity of light has 
been determined many times, with increasing accuracy. 
In our own century a highly refined technique was 


devised for this purpose by Michelson. The result of 
these experiments can be expressed simply : The velocity 
of light in vacua is approximately 186,000 miles per 
second, or 300,000 kilometres per second. 


Again we start with a few experimental facts. The 
number just quoted concerns the velocity of light in 
vacuo. Undisturbed, light travels with this speed 
through empty space. We can see through an empty 
glass vessel if we extract the air from it. We see planets, 
stars, nebulae, although the light travels from them to 
our eyes through empty space. The simple fact that we 
can see through a vessel whether or not there is air 
inside shows us that the presence of air matters very 
little. For this reason we can perform optical experi- 
ments in an ordinary room with the same effect as if 
there were no air. 

One of the simplest optical facts is that the propaga- 
tion of light is rectilinear. We shall describe a primitive 
and naive experiment showing this. In front of a point 
source is placed a screen with a hole in it. A point 
source is a very small source of light, say, a small open- 
ing in a closed lantern. On a distant wall the hole in 
the screen will be represented as light on a dark back- 
ground. The next drawing shows how this phenomenon 
is connected with the rectilinear propagation of light. 
All such phenomena, even the more complicated cases 
in which light, shadow, and half-shadows appear, can 


be explained by the assumption that light, in vacua or 
in air, travels along straight lines. 

Let us take another example, a case in which light 
passes through matter. We have a light beam travelling 
through a vacuum and falling on a glass plate. What 

happens? If the law of rectilinear motion were still 
valid, the path would be that shown by the dotted line. 
But actually it is not. There is a break in the path, such 
as is shown in the drawing. What we observe here is the 
phenomenon known as refraction. The familiar appear- 


ance of a stick which seems to be bent in the middle if 
half-immersed in water is one of the many manifestations 
of refraction. 

These facts are sufficient to indicate how a simple 
mechanical theory of light could be devised. Our aim 
here is to show how the ideas of substances, particles, 
and forces penetrated the field of optics, and how finally 
the old philosophical point of view broke down. 

The theory here suggests itself in its simplest and 
most primitive form. Let us assume that all lighted 
bodies emit particles of light, or corpuscles, which, falling 
on our eyes, create the sensation of light. We are already 
so accustomed to introduce new substances, if necessary 
for a mechanical explanation, that we can do it once 
more without any great hesitation. These corpuscles 
must travel along straight lines through empty space 
with a known speed, bringing to our eyes messages from 
the bodies emitting light. All phenomena exhibiting 
the rectilinear propagation of light support the corpus- 
cular theory, for just this kind of motion was prescribed 
for the corpuscles. The theory also explains very simply 
the reflection of light by mirrors as the same kind of 
reflection as that shown in the mechanical experiment 
of elastic balls thrown against a wall, as the next 
drawing indicates. 

The explanation of refraction is a little more difficult. 
Without going into details, we can see the possibility 
of a mechanical explanation. If corpuscles fall on the 
surface of glass, for example, it may be that a force is 
exerted on them by the particles of the matter, a force 



which strangely enough acts only in the immediate 
neighbourhood of matter. Any force acting on a 

moving particle changes the velocity, as we already 
know. If the net force on the light-corpuscles is an 
attraction perpendicular to the surface of the glass, the 
new motion will lie somewhere between the line of the 
original path and the perpendicular. This simple ex- 
planation seems to promise success for the corpuscular 
theory of light. To determine the usefulness and range 
of validity of the theory, however, we must investigate 
new and more complicated facts. 


It was again Newton's genius which explained for the 
first time the wealth of colour in the world. Here is a 
description of one of Newton's experiments in his own 
words : 

In the year 1666 (at which time I applied myself to the 
grinding of optick glasses of other figures than spherical) 
I procured me a triangular glass prism, to try therewith the 
celebrated phenomena of colours. And in order thereto, 


having darkened my chamber, and made a small hole in 
my window-shuts, to let in a convenient quantity of the 
sun's light, I placed my prism at its entrance, that it might 
thereby be refracted to the opposite wall. It was at first a 
very pleasing divertisement, to view the vivid and intense 
colours produced thereby. 

The light from the sun is "white". After passing 
through a prism it shows all the colours which exist in 
the visible world. Nature herself reproduces the same 
result in the beautiful colour scheme of the rainbow. 
Attempts to explain this phenomenon are very old. The 
Biblical story that a rainbow is God's signature to a 
covenant with man is, in a sense, a "theory". But it 
does not satisfactorily explain why the rainbow is re- 
peated from time to time, and why always in connection 
with rain. The whole puzzle of colour was first scien- 
tifically attacked and the solution pointed out in the 
great work of Newton. 

One edge of the rainbow is always red and the other 
violet. Between them all other colours are arranged. 
Here is Newton's explanation of this phenomenon: 
every colour is already present in white light. They all 
traverse interplanetary space and the atmosphere in 
unison and give the effect of white light. White light is, 
so to speak, a jnixture of corpuscies._of ^different kinds, 
belonging to different colours. In the case of Newton's 
experiment the prism separates them in space. Accord- 
ing to the mechanical theory, refraction is due to forces 
acting on the particles of light and originating from 
the particles of glass. These forces are different for 


corpuscles belonging to different colours, being strongest 
for the violet and weakest for the red. Each of the 
colours will therefore be refracted along a different path 
and be separated from the others when the light leaves 
the prism. In the case of a rainbow, drops of water play 
the role of the prism. 

The substance theory of light is now more complicated 
than before. We have not one light substance but many, 
each belonging to a different colour. If, however, there 
is some truth in the theory, its consequences must agree 
with observation. 

The series of colours in the white light of the sun, as 
revealed by Newton's experiment, is called the spectrum 
of the sun, or more precisely, its visible spectrum. The 
decomposition of white light into its components, as 
described here, is called the dispersion of light. The 
separated colours of the spectrum could be mixed 
together again by a second prism properly adjusted, 
unless the explanation given is wrong. The process 
should be just the reverse of the previous one. We 
should obtain white light from the previously separated 
colours. Newton showed by experiment that it is indeed 
possible to obtain white light from its spectrum and the 
spectrum from white light in this simple way as many 
times as one pleases. These experiments formed a strong 
support for the theory in which corpuscles belonging to 
each colour behave as unchangeable substances. Newton 
wrote thus : 

. . .which colours are not new generated, but only made 
apparent by being parted; for if they be again entirely 


mixt and blended together, they will again compose that 
colour, which they did before separation. And for the same 
reason, transmutations made by the convening of divers 
colours are not real; for when the difform rays are again 
severed, they will exhibit the very same colours which they 
did before they entered the composition; as you see blue 
and yellow powders, when finely mixed, appear to the naked 
eye, green, and yet the colours of the component corpuscles 
are not thereby really transmuted, but only blended. For 
when viewed with a good microscope they still appear blue 
and yellow interspersedly. 

Suppose that we have isolated a very narrow strip of 
the spectrum. This means that of all the many colours 
we allow only one to pass through the slit, the others 
being stopped by a screen. The beam which comes 
through will consist of homogeneous light, that is, light 
which cannot be split into further components. This is 
a consequence of the theory and can be easily con- 
firmed by experiment. In no way can such a beam of 
single colour be divided further. There are simple means 
of obtaining sources of homogeneous light. For ex- 
ample, sodium, when incandescent, emits homogeneous 
yellow light. It is very often convenient to perform 
certain optical experiments with homogeneous light, 
since, as we can well understand, the result will be 
much simpler. 

Let us imagine that suddenly a very strange thing 
happens: our sun begins to emit only homogeneous 
light of some definite colour, say yellow. The great 
variety of colours on the earth would immediately 
vanish. Everything would be either yellow or black! 


This prediction is a consequence of the substance theory 
of light, for new colours cannot be created. Its validity 
can be confirmed by experiment: in a room where the 
only source of light is incandescent sodium everything 
is either yellow or black. The wealth of colour in the 
world reflects the variety of colour of which white light 
is composed. 

The substance theory of light seems to work splendidly 
in all these cases, although the necessity for introducing 
as many substances as colours may make us somewhat 
uneasy. The assumption that all the corpuscles of light 
have exactly the same velocity in empty space also seems 
very artificial. 

It is imaginable that another set of suppositions, a 
theory of entirely different character, would work just 
as well and give all the required explanations. Indeed, 
we shall soon witness the rise of another theory, based 
on entirely different concepts, yet explaining the same 
domain of optical phenomena. Before formulating the 
underlying assumptions of this new theory, however, we 
must answer a question in no way connected with these 
optical considerations. We must go back to mechanics 
and ask: 


A bit of gossip starting in London reaches Edinburgh 
very quickly, even though not a single individual 
who takes part in spreading it travels between these 
two cities. There are two quite different motions in- 
volved, that of the rumour, London to Edinburgh, 
and that of the persons who spread the rumour. The 


wind, passing over a field of grain, sets up a wave 
which spreads our across the whole field. Here again 
we must distinguish between the motion of the wave 
and the motion of the separate plants, which undergo 
only small oscillations. We have all seen the waves that 
spread in wider and wider circles when a stone is 
thrown into a pool of water. The motion of the wave 
is very different from that of the particles of water. 
The particles merely go up and down. The observed 
motion of the wave is that of a state of matter and not 
of matter itself. A cork floating on the wave shows this 
clearly, for it moves up and down in imitation of the 
actual motion of the water, instead of being carried 
along by the wave. 

In order to understand better the mechanism of the 
wave let us again consider an idealized experiment. 
Suppose that a large space is filled quite uniformly with 
water, or air, or some other "medium". Somewhere in 
the centre there is a sphere. At the beginning of the 
experiment there is no motion at all. Suddenly the 
sphere begins to "breathe" rhythmically, expanding 
and contracting in volume, although retaining its 
spherical shape. What will happen in the medium? Let 
us begin our examination at the moment the sphere 
begins to expand. The particles of the medium in the 
immediate vicinity of the sphere are pushed out, so that 
the density of a spherical shell of water, or air, as the case 
may be, is increased above its normal value. Similarly, 
when the sphere contracts, the density of that part 
of the medium immediately surrounding it will be 


decreased. These changes of density are propagated 
throughout the entire medium. The particles constitut- 
ing the medium perform only small vibrations, but 
the whole motion is that of a progressive wave. The 
essentially new thing here is that for the first time we 
consider the motion of something which is not matter, 
but energy propagated through matter. 

Using the example of the pulsating sphere, we may 
introduce two general physical concepts, important for 
the characterization of waves. The first is the velocity 
with which the wave spreads. This will depend on the 
medium, being different for water and air, for example. 
The second concept is that of wave-length. In the case 
of waves on a sea or river it is the distance from the 
trough of one wave to that of the next, or from the 
crest of one wave to that of the next. Thus sea waves 
have greater wave-length than river waves. In the 
case of our waves set up by a pulsating sphere the 
wave-length is the distance, at some definite time, 
between two neighbouring spherical shells showing 
maxima or minima of density. It is evident that this 
distance will not depend on the medium alone. The rate 
of pulsation of the sphere will certainly have a great 
effect, making the wave-length shorter if the pulsation 
becomes more rapid, longer if the pulsation becomes 

This concept of a wave proved very successful in 
physics. It is definitely a mechanical concept. The 
phenomenon is reduced to the motion of particles which, 
according to the kinetic theory, are constituents of 


matter. Thus every theory which uses the concept of 
wave can, in general, be regarded as a mechanical 
theory. For example, the explanation of acoustical 
phenomena is based essentially on this concept. Vibrat- 
ing bodies, such as vocal cords and violin strings, are 
sources of sound waves which are propagated through 
the air in the manner explained for the pulsating sphere. 
It is thus possible to reduce all acoustical phenomena to 
mechanics by means of the wave concept. 

It has been emphasized that we must distinguish 
between the motion of the particles and that of the wave 
itself, which is a state of the medium. The two are 
very different, but it is apparent that in our example of 
the pulsating sphere both motions take place in the 

same straight line. The particles of the medium oscillate 
along short line segments, and the density increases and 
decreases periodically in accordance with this motion. 


The direction in which the wave spreads and the line 
on which the oscillations lie are the same. This type of 
wave is called longitudinal. But is this the only kind of 
wave? It is important for our further considerations to 
realize the possibility of a different kind of wave, called 

Let us change our previous example. We still have 
the sphere, but it is immersed in a medium of a different 
kind, a sort of jelly instead of air or water. Further- 
more, the sphere no longer pulsates but rotates in one 
direction through a small angle and then back again, 

always in the same rhythmical way and about a definite 
axis. The jelly adheres to the sphere and thus the 
adhering portions are forced to imitate the motion. 
These portions force those situated a little farther away 
to imitate the same motion, and so on, so that a wave is 
set up in the medium. If we keep in mind the distinction 


between the motion of the medium and the motion 
of the wave, we see that here they do not lie on the same 
line. The wave is propagated in the direction of the 
radius of the sphere, while the parts of the medium 
move perpendicularly to this direction. We have thus 
created a transverse wave. 

Waves spreading on the surface of water are trans- 
verse. A floating cork only bobs up and down, but the 
wave spreads along a horizontal plane. Sound waves, 
on the other hand, furnish the most familiar example of 
longitudinal waves. ^ 

One more remark : the wave produced by a pulsating 
or oscillating sphere in a homogeneous medium is a 
spherical wave. It is called so because at any given 
moment all points on any sphere surrounding the 
source behave in the same way// Let us consider a 
portion of such a sphere at a great distance from the 
source. The farther away the portion is, and the smaller 
we choose to take it, the more it resembles a plane. We 
can say, without trying to be too rigorous, that there 
is no essential difference between a part of a plane and 


a part of a sphere whose radius is sufficiently large. 
We very often speak of small portions of a spherical 
wave far removed from the source as plane waves. The 
farther we place the shaded portion of our drawing 
from the centre of the spheres and the smaller the angle 
between the two radii, the better our representation of 
a plane wave. The concept of a plane wave, like many 
other physical concepts, is no more than a fiction which 
can be realized with only a certain degree of accuracy. 
It is, however, a useful concept which we shall need 


Let us recall why we broke off the description of 
optical phenomena. Our aim was to introduce another 
theory of light, different from the corpuscular one, but 
also attempting to explain the same domain of facts. 
To do this we had to interrupt our story and introduce 
the concept of waves. Now we can return to our 

It was Huygens, a contemporary of Newton, who put 
forward quite a new theory. In his treatise on light he 
wrote : 

If, in addition, light takes time for its passage which we 
are now going to examine it will follow that this move- 
ment, impressed on the intervening matter, is successive; 
and consequently it spreads, as sound does, by spherical 
surfaces and waves, for I call them waves from their resem- 
blance to those which are seen to be formed in water when 
a stone is thrown into it, and which present a successive 


spreading as circles, though these arise from another cause, 
and are only in a flat surface. 

According to Huygens, light is a wave, a transference 
of energy and not of substance. We have seen that the 
corpuscular theory explains many of the observed facts. 
Is the wave theory also able to do this? We must 
again ask the questions which have already been 
answered by the corpuscular theory, to see whether 
the wave theory can do the answering just as well. We 
shall do this here in the form of a dialogue between N 
and //, where N is a believer in Newton's corpuscular 
theory, and H in Huygen's theory. Neither is allowed 
to use arguments developed after the work of the two 
great masters was finished. 

JV. In the corpuscular theory the velocity of light has 
a very definite meaning. It is the velocity at which the 
corpuscles travel through empty space. What does it 
mean in the wave theory? 

H. It means the velocity of the light wave, of course. 
Every known wave spreads with some definite velocity, 
and so should a wave of light. 

N. That is not as simple as it seems. Sound waves 
spread in air, ocean waves in water. Every wave must 
have a material medium in which it travels. But light 
passes through a vacuum, whereas sound does not. To 
assume a wave in empty space really means not to 
assume any wave at all. 

H. Yes, that is a difficulty, although not a new one 
to me. My master thought about it very carefully, and 
decided that the only way out is to assume the existence 


of a hypothetical substance, the ether, a transparent 
medium permeating the entire universe. The universe 
is, so to speak, immersed in ether. Once we have the 
courage to introduce this concept, everything else 
becomes clear and convincing. 

JV. But I object to such an assumption. In the first 
place it introduces a new hypothetical substance, and 
we already have too many substances in physics. There 
is also another reason against it. You no doubt believe 
that we must explain everything in terms of mechanics. 
But what about the ether? Are you able to answer the 
simple question as to how the ether is constructed from 
its elementary particles and how it reveals itself in other 

//. Your first objection is certainly justified. But by 
introducing the somewhat artificial weightless ether 
we at once get rid of the much more artificial light 
corpuscles. We have only one "mysterious" substance 
instead of an infinite number of them corresponding to 
the great number of colours in the spectrum. Do you 
not think that this is real progress? At least all the 
difficulties are concentrated on one point. We no 
longer need the factitious assumption that particles 
belonging to different colours travel with the same 
speed through empty space. Your second argument is 
also true. We cannot give a mechanical explanation of 
ether. But there is no doubt that the future study of 
optical and perhaps other phenomena will reveal its 
structure. At present we must wait for new experi- 
ments and conclusions, but finally, I hope, we shall be 


able to clear up the problem of the mechanical structure 
of the ether. 

JV. Let us leave the question for the moment, since 
it cannot be settled now. I should like to see how your 
theory, even if we waive the difficulties, explains those 
phenomena which are so clear and understandable in 
the light of the corpuscular theory. Take, for example, 
the fact that light rays travel in vacuo or in air along 
straight lines. A piece of paper placed in front of a 
candle produces a distinct and sharply outlined shadow 
on the wall. Sharp shadows would not be possible if 
the wave theory of light were correct, for waves would 
bend around the edges of the paper and thus blur the 
shadow. A small ship is not an obstacle for waves on 
the sea, you know; they simply bend around it without 
casting a shadow. 

H. That is not a convincing argument. Take short 
waves on a river impinging on the side of a large ship. 
Waves originating on one side of the ship will not be 
seen on the other. If the waves are small enough and 
the ship large enough, a very distinct shadow appears. 
It is very probable that light seems to travel in straight 
lines only because its wave-length is very small in 
comparison with the size of ordinary obstacles and of 
apertures used in experiments. Possibly, if we could 
create a sufficiently small obstruction, no shadow would 
occur. We might meet with great experimental diffi- 
culties in constructing apparatus which would show 
whether light is capable of bending. Nevertheless, if 
such an experiment could be devised it would be crucial 


in deciding between the wave theory and the corpuscular 
theory of light. 

JV". The wave theory may lead to new facts in the 
future, but I do not know of any experimental data 
confirming it convincingly. Until it is definitely proved 
by experiment that light may be bent, I do not see any 
reason for not believing in the corpuscular theory, 
which seems to me to be simpler, and therefore better, 
than the wave theory. 

At this point we may interrupt the dialogue, though 
the subject is by no means exhausted. 

It still remains to be shown how the wave theory 
explains the refraction of light and the variety of colours. 
The corpuscular theory is capable of this, as we know. 
We shall begin with refraction, but it will be useful to 
consider first an example having nothing to do with 

There is a large open space in which there are 
walking two men holding between them a rigid pole. 
At the beginning they are walking straight ahead, both 
with the same velocity. As long as their velocities re- 
main the same, whether great or small, the stick will 
be undergoing parallel displacement; that is, it does not 
turn or change its direction. All consecutive positions 
of the pole are parallel to each other. But now imagine 
that for a time which may be as short as a fraction of 
a second the motions of the two men are not the same. 
What will happen? It is clear that during this moment 
the stick will turn, so that it will no longer be displaced 
parallel to its original position. When the equal velocities 


are resumed, it is in a direction different from the 
previous one. This is shown clearly in the drawing. 

The change in direction took place during the time 
interval in which the velocities of the two walkers were 

This example will enable us to understand the re- 
fraction of a wave. A plane wave travelling through 
the ether strikes a plate of glass. In the next drawing 
we see a wave which presents a comparatively wide front 
as it marches along. The wave front is a plane on which 
at any given moment all parts of the ether behave in 
precisely the same way. Since the velocity depends on 
the medium through which the light is passing, it will 
be different in glass from the velocity in empty space. 
During the very short time in which the wave front 
enters the glass, different parts of the wave front will 
have different velocities. It is clear that the part which 
has reached the glass will travel with the velocity of light 
in glass, while the other still moves with the velocity 
of light in ether. Because of this difference in velocity 



along the wave front during the time of "immersion" 
in the glass, the direction of the wave itself will be 

Thus we see that not only the corpuscular theory, 
but also the wave theory, leads to an explanation of 
refraction. Further consideration, together with a little 
mathematics, shows that the wave theory explanation 
is simpler and better, and that the consequences are in 
perfect agreement with observation. Indeed, quanti- 
tative methods of reasoning enable us to deduce the 
velocity of light in a refractive medium if we know how 
the beam refracts when passing into it. Direct measure- 
ments splendidly confirm these predictions, and thus 
also the wave theory of light. 

There still remains the question of colour. 

It must be remembered that a wave is characterized 
by two numbers, its velocity and its wave-length. The 
essential assumption of the wave theory of light is that 
different wave-lengths correspond to different colours. The 


wave-length of homogeneous yellow light differs from 
that of red or violet. Instead of the artificial segregation 
of corpuscles belonging to various colours we have the 
natural difference in wave-length. 

It follows that Newton's experiments on the dispersion 
of light can be described in two different languages, that 
of the corpuscular theory and that of the wave theory. 
For example : 


The corpuscles belonging The rays of different wave 
to different colours have the length belonging to differ- 
same velocity in vacuo, but ent colours have the same 
different velocities in glass. velocity in the ether, but 

different velocities in glass. 

White light is a composi- White light is a composi- 
tion of corpuscles belonging tion of waves of all wave- 
to different colours, whereas lengths, whereas in the spec- 
in the spectrum they are trum they are separated, 

It would seem wise to avoid the ambiguity resulting 
from the existence of two distinct theories of the same 
phenomena, by deciding in favour of one of them after 
a careful consideration of the faults and merits of each. 
The dialogue between JV and H shows that this is no 
easy task. The decision at this point would be more a 
matter of taste than of scientific conviction. In Newton's 
time, and for more than a hundred years after, most 
physicists favoured the corpuscular theory. 

History brought in its verdict, in favour of the wave 
theory of light and against the corpuscular theory, at 


a much later date, the middle of the nineteenth 
century. In his conversation with //, JV stated that a 
decision between the two theories was, in principle, 
experimentally possible. The corpuscular theory does 
not allow light to bend, and demands the existence of 
sharp shadows. According to the wave theory, on the 
other hand, a sufficiently small obstacle will cast no 
shadow. In the work of Young and Fresnel this result 
was experimentally realized and theoretical conclusions 
were drawn. 

An extremely simple experiment has already been 
discussed, in which a screen with a hole was placed in 
front of a point source of light and a shadow appeared 
on the wall. We shall simplify the experiment further 
by assuming that the source emits homogeneous light. 
For the best results the source should be a strong one. 
Let us imagine that the hole in the screen is made 
smaller and smaller. If we use a strong source and 
succeed in making the hole small enough, a new and 
surprising phenomenon appears, something quite in- 
comprehensible from the point of view of the corpus- 
cular theory. There is no longer a sharp distinction 
between light and dark. Light gradually fades into the 
dark background in a series of light and dark rings. 
The appearance of rings is very characteristic of a 
wave theory. The explanation for alternating light and 
dark areas will be clear in the case of a somewhat 
different experimental arrangement. Suppose we have 
a sheet of dark paper with two pinholes through which 
light may pass. If the holes are close together and very 


(Photographed by V. Arkadiev) 

Above, we see a photograph of light spots 
after two beams have passed through two 
pin holes, one after the other. (One pin hole 
was opened, then covered and the other 
opened.) Below, we sec stripes when light 
is allowed to pass through both pin holes 

(Photographed by I'. Arkadiev) 

Diffraction of light bending 
around a small obstacle 

Diffraction of light passing 
through a small hole 


small, and if the source of homogeneous light is strong 
enough, many light and dark bands will appear on the 
wall, gradually fading off at the sides into the dark 
background. The explanation is simple. "A dark band 
is where a trough of a wave from one pinhole meets the 
crest of a wave from the other pinhole, so that the two 
cancel. A band of light is where two troughs or two 
crests from waves of the different pinholes meet and 
reinforce each other,/ The explanation is more com- 
plicated in the case of the dark and light rings of our 
previous example in which we used a screen with one 
hole, but the principle is the same. This appearance of 
dark and light stripes in the case of two holes and of 
light and dark rings in the case of one hole should be 
borne in mind, for we shall later return to a discussion 
of the two different pictures. The experiments described 
here show the diffraction of light, the deviation from the 
rectilinear propagation when small holes or obstacles 
are placed in the way of the light wave. 

With the aid of a little mathematics we are able to 
go much further. It is possible to find out how great 
or, rather, how small the wave-length must be to pro- 
duce a particular pattern. Thus the experiments de- 
scribed enable us to measure the wave-length of the 
homogeneous light used as a source. To give an idea of 
how small the numbers are we shall cite two wave- 
lengths, those representing the extremes of the solar 
spectrum, that is, the red and the violet. 

The wave-length of red light is 0-00008 cm. 
The wave-length of violet light is 0-00004 cm. 


We should not be astonished that the numbers are 
so small. The phenomenon of distinct shadow, that is, 
the phenomenon of rectilinear propagation of light, is 
observed in nature only because all apertures and 
obstacles ordinarily met with are extremely large in 
comparison with the wave-lengths of light. It is only 
when very small obstacles and apertures are used that 
light reveals its wave-like nature. 

But the story of the search for a theory of light is by 
no means finished. The verdict of the nineteenth century 
was not final and ultimate. For the modern physicist 
the entire problem of deciding between corpuscles and 
waves again exists, this time in a much more profound 
and intricate form. Let us accept the defeat of the 
corpuscular theory of light until we recognize the 
problematic nature of the victory of the wave theory. 


All the optical phenomena we have considered speak 
for the wave theory. The bending of light around small 
obstacles and the explanation of refraction are the 
strongest arguments in its favour. Guided by the 
mechanical point of view we realize that there is still 
one question to be answered : the determination of the 
mechanical properties of the ether. It is essential for 
the solution of this problem to know whether light waves 
in the ether are longitudinal or transverse. In other 
words: is light propagated like sound? Is the wave due 
to changes in the density of the medium, so that the 
oscillations of the particles are in the direction of the 


propagation? Or does the ether resemble an elastic 
jelly, a medium in which only transverse waves can be 
set up and whose particles move in a direction per- 
pendicular to that in which the wave itself travels? 

Before solving this problem, let us try to decide 
which answer should be preferred. Obviously, we 
should be fortunate if light waves were longitudinal. 
The difficulties in designing a mechanical ether would 
be much simpler in this case. Our picture of ether 
might very probably be something like the mechanical 
picture of a gas that explains the propagation of sound 
waves. It would be much more difficult to form a 
picture of ether carrying transverse waves. To imagine 
a jelly as a medium made up of particles in such a way 
that transverse waves are propagated by means of it is 
no easy task. Huygens believed that the ether would 
turn out to be " air-like" rather than "jelly-like". But 
nature cares very little for our limitations. Was nature, 
in this case, merciful to the physicists attempting to 
understand all events from a mechanical point of view? 
In order to answer this question we must discuss some 
new experiments. 

We shall consider in detail only one of many ex- 
periments which are able to supply us with an answer. 
Suppose we have a very thin plate of tourmaline 
crystal, cut in a particular way which we need not 
describe here. The crystal plate must be thin so that 
we are able to see a source of light through it. But now 
let us take two such plates and place both of them 
between our eyes and the light. What do we expect to 


see? Again a point of light, if the plates are sufficiently 
thin. The chances are very good that the experiment 
will confirm our expectation. Without worrying about 
the statement that it may be chance, let us assume we 
do see the light point through the two crystals. Now 
let us gradually change the position of one of the 
crystals by rotating it. This statement makes sense only 
if the position of the axis about which the rotation 
takes place is fixed. We shall take as an axis the line 
determined by the incoming ray. This means that we 
displace all the points of the one crystal except those 

on the axis. A strange thing happens ! The light gets 
weaker and weaker until it vanishes completely. It 
reappears as the rotation continues and we regain the 
initial view when the initial position is reached. 

Without going into the details of this and similar 


experiments we can ask the following question: can 
these phenomena be explained if the light waves are 
longitudinal? In the case of longitudinal waves the 
particles of the ether would move along the axis, as the 
beam does. If the crystal rotates, nothing along the axis 
changes. The points on the axis do not move, and only 
a very small displacement takes place nearby. No such 
distinct change as the vanishing and appearance of a 
new picture could possibly occur for a longitudinal 
wave. This and many other similar phenomena can be 
explained only by the assumption that light waves are 
transverse and not longitudinal ! Or, in other words, 
the " jelly-like" character of the ether must be assumed. 
This is very sad ! We must be prepared to face 
tremendous difficulties in the attempt to describe the 
ether mechanically. 

The discussion of all the various attempts to understand 
the mechanical nature of the ether as a medium for 
transmitting light would make a long story. A me- 
chanical construction means, as we know, that the 
substance is built up of particles with forces acting 
along lines connecting them and depending only on the 
distance. In order to construct the ether as a jelly-like 
mechanical substance physicists had to make some 
highly artificial and unnatural assumptions. We shall 
not quote them here; they belong to the almost for- 
gotten past. But the result was significant and impor- 
tant. The artificial character of all these assumptions, 


the necessity for introducing so many of them all quite 
independent of each other, was enough to shatter the 
belief in the mechanical point of view. 

But there are other and simpler objections to ether 
than the difficulty of constructing it. Ether must be 
assumed to exist everywhere, if we wish to explain 
optical phenomena mechanically. There can be no 
empty space if light travels only in a medium. 

Yet we know from mechanics that interstellar space 
does not resist the motion of material bodies. The 
planets, for example, travel through the ether-jelly 
without encountering any resistance such as a material 
medium would offer to their motion. If ether does not 
disturb matter in its motion, there can be no interaction 
between particles of ether and particles of matter. Light 
passes through ether and also through glass and water, 
but its velocity is changed in the latter substances. How 
can this fact be explained mechanically? Apparently 
only by assuming some interaction between ether 
particles and matter particles. We have just seen that 
in the case of freely moving bodies such interactions 
must be assumed not to exist. In other words, there is 
interaction between ether and matter in optical pheno- 
mena, but none in mechanical phenomena ! This is 
certainly a very paradoxical conclusion ! 

There seems to be only one way out of all these 
difficulties. In the attempt to understand the pheno- 
mena of nature from the mechanical point of view, 
throughout the whole development of science up to 
the twentieth century, it was necessary to introduce 


artificial substances like electric and magnetic fluids, 
light corpuscles, or ether. The result was merely the 
concentration of all the difficulties in a few essential 
points, such as ether in the case of optical phenomena. 
Here all the fruitless attempts to construct an ether in 
some simple way, as well as the other objections, seem 
to indicate that the fault lies in the fundamental assump- 
tion that it is possible to explain all events in nature from 
a mechanical point of view. Science did not succeed in 
carrying out the mechanical programme convincingly, 
and today no physicist believes in the possibility of its 

In our short review of the principal physical ideas 
we have met some unsolved problems, have come 
upon difficulties and obstacles which discouraged the 
attempts to formulate a uniform and consistent view of 
all the phenomena of the external world. There was 
the unnoticed clue in classical mechanics of the equality 
of gravitational and inertial mass. There was the 
artificial character of the electric and magnetic fluids. 
There was, in the interaction between electric current 
and magnetic needle, an unsolved difficulty. It will be 
remembered that this force did not act in the line con- 
necting the wire and the magnetic pole, and depended 
on the velocity of the moving charge. The law expressing 
its direction and magnitude was extremely complicated. 
And finally, there was the great difficulty with the 

Modern physics has attacked all these problems and 
solved them. But in the struggle for these solutions 


new and deeper problems have been created. Our 
knowledge is now wider and more profound than that 
of the physicist of the nineteenth century, but so are 
our doubts and difficulties. 


In the old theories of electric fluids, in the corpuscular 
and wave theories of light, we witness the further attempts 
to apply the mechanical view. But in the realm of electric 
and optical phenomena we meet grave difficulties in this 

A moving charge acts upon a magnetic needle. But the force, 
instead of depending only upon distance, depends also upon the 
velocity of the charge. The force neither repels not attracts but 
acts perpendicular to the line connecting the needle and the 

In optics we have to decide in favour of the wave theory 
against the corpuscular theory of light. Waves spreading in a 
medium consisting of particles, with mechanical forces acting 
between them, are certainly a mechanical concept. But what is 
the medium through which light spreads and what are its 
mechanical properties? There is no hope of reducing the 
optical phenomena to the mechanical ones before this question 
is answered. But the difficulties in solving this problem are so 
great that we have to give it up and thus give up the mechanical 
views as well. 



The field as representation The two pillars of the field 
theory The reality of the field Field and ether The 
mechanical scaffold Ether and motion Time, distance, re- 
lativity Relativity and mechanics The time-space con- 
tinuum General relativity Outside and inside the lift 
Geometry and experiment General relativity and its verifi- 
cation Field and matter 

DURING the second half of the nineteenth century new 
and revolutionary ideas were introduced into physics; 
they opened the way to a new philosophical view, 
differing from the mechanical one. The results of the 
work of Faraday, Maxwell, and Hertz led to the de- 
velopment of modern physics, to the creation of new 
concepts, forming a new picture of reality. 

Our task now is to describe the break brought about 
in science by these new concepts and to show how 
they gradually gained clarity and strength. We shall 
try to reconstruct the line of progress logically, without 
bothering too much about chronological order. 

The new concepts originated in connection with the 
phenomena of electricity, but it is simpler to introduce 
them, for the first time, through mechanics. We know 
that two particles attract each other and that this force 
of attraction decreases with the square of the distance. 
EE 129 5 


We can represent this fact in a new way, and shall do 
so even though it is difficult to understand the advan- 
tage of this. The small circle in our drawing represents 

an attracting body, say, the sun. Actually, our diagram 
should be imagined as a model in space and not as a 
drawing on a plane. Our small circle, then, stands for 
a sphere in space, say, the sun. A body, the so-called 
test body, brought somewhere within the vicinity of 
the sun will be attracted along the line connecting the 
centres of the two bodies. Thus the lines in our draw- 
ing indicate the direction of the attracting force of the 
sun for different positions of the test body. The arrow 
on each line shows that the force is directed toward 
the sun; this means the force is an attraction. These are 
the lines of force of the gravitational field. For the moment, 
this is merely a name and there is no reason for stress- 
ing it further. There is one characteristic feature of 
our drawing which will be emphasized later. The lines 


of force are constructed in space, where no matter is 
present. For the moment, all the lines of force, or 
briefly speaking, the field, indicate only how a test body 
would behave if brought into the vicinity of the sphere 
for which the field is constructed. 

The lines in our space model are always perpen- 
dicular to the surface of the sphere. Since they diverge 
from one point, they are dense near the sphere and 
become less and less so farther away. If we increase the 
distance from the sphere twice or three times, then the 
density of the lines, in our space model, though not in 
the drawing, will be four or nine times less. Thus the 
lines serve a double purpose. On the one hand, they 
show the direction of the force acting on a body 
brought into the neighbourhood of the sphere-sun. On 
the other hand, the density of the lines of force in space 
shows how the force varies with the distance. The 
drawing of the field, correctly interpreted, represents 
the direction of the gravitational force and its depend- 
ence on distance. One can read the law of gravitation 
from such a drawing just as well as from a description 
of the action in words, or in the precise and econo- 
mical language of mathematics. This field representation, 
as we shall call it, may appear clear and interesting, but 
there is no reason to believe that it marks any real ad- 
vance. It would be quite difficult to prove its usefulness 
in the case of gravitation. Some may, perhaps, find it 
helpful to regard these lines as something more than 
drawings, and to imagine the real actions of force pass- 
ing through them. This may be done, but then the 



speed of the actions along the lines of force must be 
assumed as infinitely great ! The force between two 
bodies, according to Newton's law, depends only on 
distance ; time does not enter the picture. The force has 
to pass from one body to another in no time ! But, as 
motion with infinite speed cannot mean much to any 
reasonable person, an attempt to make our drawing 
something more than a model leads nowhere. 

We do not intend, however, to discuss the gravita- 
tional problem just now It served only as an introduc- 
tion, simplifying the explanation of similar methods of 
reasoning in the theory of electricity. 

We shall begin with a discussion of the experiment 
which created serious difficulties in our mechanical 
interpretation. We had a current flowing through a 
wire circuit in the form of a circle. In the middle of the 
circuit was a magnetic needle. The moment the current 
began to flow a new force appeared, acting on the 
magnetic pole, and perpendicular to any line connect- 
ing the wire and the pole. This force, if caused by a 
circulating charge, depended, as shown by Rowland's 
experiment, on the velocity of the charge. These experi- 
mental facts contradicted the philosophical view that 
all forces must act on the line connecting the particles 
and can depend only upon distance. 

The exact expression for the force of a current act- 
ing on a magnetic pole is quite complicated, much 
more so, indeed, than the expression for gravitational 
forces. We can, however, attempt to visualize the ac- 
tions just as we did in the case of a gravitational force. 


Our question is: with what force does the current act 
upon a magnetic pole placed somewhere in its vicin- 
ity? It would be rather difficult to describe this force 
in words. Even a mathematical formula would be 
complicated and awkward. It is best to represent all 
we know about the acting forces by a drawing, or 
rather by a spatial model, with lines of force. Some 
difficulty is caused by the fact that a magnetic pole 
exists only in connection with another magnetic pole, 
forming a dipole. We can, however, always imagine 
the magnetic needle of such length that only the force 
acting upon the pole nearer the current has to be taken 
into account. The other pole is far enough away for 
the force acting upon it to be negligible. To avoid 
ambiguity we shall say that the magnetic pole brought 
nearer to the wire is the positive one. 

The character of the force acting upon the positive 
magnetic pole can be read from our drawing. 

First we notice an arrow near the wire indicating 
the direction of the current, from higher to lower 


potential. All other lines are just lines offeree belonging 
to this current and lying on a certain plane. If drawn 
properly, they tell us the direction of the force vector 
representing the action of the current on a given posi- 
tive magnetic pole as well as something about the 
length of this vector. Force, as we know, is a vector, 
and to determine it we must know its direction as well 
as its length. We are chiefly concerned with the pro- 
blem of the direction of the force acting upon a pole. 
Our question is: how can we find, from the drawing, 
the direction of the force, at any point in space? 

The rule for reading the direction of a force from 
such a model is not as simple as in our previous ex- 
ample, where the lines of force were straight. In our 
next diagram only one line of force is drawn in order 

to clarify the procedure. The force vector lies on the 
tangent to the line of force, as indicated. The arrow of 
the force vector and the arrows on the line of force 
point in the same direction. Thus this is the direction 
in which the force acts on a magnetic pole at this 


point. A good drawing, or rather a good model, also 
tells us something about the length of the force vector 
at any point. This vector has to be longer where the 
lines are denser, i.e., near the wire, shorter where the 
lines are less dense, i.e., far from the wire. 

In this way, the lines of force, or in other words, the 
field, enable us to determine the forces acting on a 
magnetic pole at any point in space. This, for the 
time being, is the only justification for our elaborate 
construction of the field. Knowing what the field ex- 
presses, we shall examine with a far deeper interest the 
lines of force corresponding to the current. These lines 
are circles surrounding the wire and lying on the plane 
perpendicular to that in which the wire is situated. 
Reading the character of the force from the drawing, 
we come once more to the conclusion that the force 
acts in a direction perpendicular to any line connecting 
the wire and the pole, for the tangent to a circle is 
always perpendicular to its radius. Our entire know- 
ledge of the acting forces can be summarized in the 
construction of the field. We sandwich the concept of 
the field between that of the current and that of the 
magnetic pole in order to represent the acting forces 
in a simple way. 

Every current is associated with a magnetic field, 
i.e., a force always acts on a magnetic pole brought 
near the wire through which a current flows. We may 
remark in passing that this property enables us to con- 
struct sensitive apparatus for detecting the existence of 
a current. Once having learned how to read the charac- 


ter of the magnetic forces from the field model of a 
current, we shall always draw the field surrounding 
the wire through which the current flows, in order to 
represent the action of the magnetic forces at any 
point in space. Our first example is the so-called 
solenoid. This is, in fact , a coil of wire as shown in 
the drawing. Our aim is to learn, by experiment, all 
we can about the magnetic field associated with the 
current flowing through a solenoid and to incorporate 
this knowledge in the construction of a field. A drawing 

represents our result. The curved lines of force are 
closed, and surround the solenoid in a way character- 
istic of the magnetic field of a current. 

The field of a bar magnet can be represented in the 
same way as that of a current. Another drawing shows 
this. The lines of force are directed from the positive 
to the negative pole. The force vector always lies on^ 
the tangent to the line of force and is longest near the 
poles because the density of the lines is greatest at these 
points. The force vector represents the action of the 


magnet on a positive magnetic pole. In this case the 
magnet and not the current is the " source" of the field. 

Our last two drawings should be carefully compared. 
In the first, we have the magnetic field of a current 
flowing through a solenoid; in the second, the field 
of a bar magnet. Let us ignore both the solenoid and 
the bar and observe only the two outside fields. We 
immediately notice that they are of exactly the same 
character; in each case the lines offeree lead from one 
end of the solenoid or bar to the other. 

The field representation yields its first fruit ! It 
would be rather difficult to see any strong similarity 
between the current flowing through a solenoid and a 
bar magnet if this were not revealed by our construction 
of the field. 

The concept of field can now be put to a much 
more severe test. We shall soon see whether it is any- 
thing more than a new representation of the acting 
forces. We could reason: assume, for a moment, that 
the field characterizes all actions determined by its 
sources in a unique way. This is only a guess. It would 
mean that if a solenoid and a bar magnet have the same 


field, then all their influences must also be the same. 
It would mean that two solenoids, carrying electric 
currents, behave like two bar magnets, that they attract 
or repel each other, depending exactly as in the case of 
bars, on their relative positions. It would also mean that 
a solenoid and a bar attract or repel each other in the 
same way as two bars. Briefly speaking, it would mean 
that all actions of a solenoid through which a current 
flows and of a corresponding bar magnet are the same, 
since the field alone is responsible for them, and the 
field in both cases is of the same character. Experiment 
fully confirms our guess ! 

How difficult it would be to find those facts without 
the concept of field ! The expression for a force acting 
between a wire through which a current flows and a 
magnetic pole is very complicated. In the case of two 
solenoids, we should have to investigate the forces with 
which two currents act upon each other. But if we do 
this, with the help of the field, we immediately notice 
the character of all those actions at the moment when 
the similarity between the field of a solenoid and that 
of a bar magnet is seen. 

We have the right to regard the field as something 
much more than we did at first. The properties of the 
field alone appear to be essential for the description of 
phenomena; the differences in source do not matter. 
The concept of field reveals its importance by leading 
to new experimental facts. 

The field proved a very helpful concept. It began as 
something placed between the source and the magnetic 


needle in order to describe the acting force. It was 
thought of as an "agent" of the current, through which 
all action of the current was performed. But now the 
agent also acts as an interpreter, one who translates the 
laws into a simple, clear language, easily understood. 

The first success of the field description suggests 
that it may be convenient to consider all actions of 
currents, magnets and charges indirectly, i.e., with the 
help of the field as an interpreter. A field may be re- 
garded as something always associated with a current. 
It is there even in the absence of a magnetic pole to 
test its existence. Let us try to follow this new clue 

The field of a charged conductor can be introduced 
in much the same way as the gravitational field, or the 
field of a current or magnet. Again only the simplest 
example ! To design the field of a positively charged 
sphere, we must ask what kind of forces are acting on 



a small positively charged test body brought near the 
source of the field, the charged sphere. The fact that 
we use a positively and not a negatively charged test 
body is merely a convention, indicating in which direc- 
tion the arrows on the line of force should be drawn. 
The model is analogous to that of a gravitational field 
(p. 130) because of the similarity between Coulomb's 
law and Newton's. The only difference between the 
two models is that the arrows point in opposite direc- 
tions. Indeed, we have repulsion of two positive 
charges and attraction of two masses. However, the 
field of a sphere with a negative charge will be iden- 
tical with a gravitational field since the small positive 
testing charge will be attracted by the source of the 

If both electric and magnetic poles are at rest, there 
is no action between them, neither attraction nor re- 
pulsion. Expressing the same fact in the field language, 


we can say: an electrostatic field does not influence a 
magnetostatic one and vice versa. The words "static 
field" mean a field that does not change with time. 
The magnets and charges would rest near one another 
for an eternity if no external forces disturbed them. 
Electrostatic, magnetostatic and gravitational fields are 
all of different character. They do not mix; each pre- 
serves its individuality regardless of the others. 

Let us return to the electric sphere which was, until 
now, at rest, and assume that it begins to move owing 
to the action of some external force. The charged sphere 
moves. In the field language this sentence reads: the 
field of the electric charge changes with time. But the 
motion of this charged sphere is, as we already know 
from Rowland's experiment, equivalent to a current. 
Further, every current is accompanied by a magnetic 
field. Thus the chain of our argument is : 

motion of charge -> change of an electric field 

current -> associated magnetic field. 

We, therefore, conclude : The change of an electric field 
produced by the motion of a charge is always accompanied by 
a magnetic fold. 

Our conclusion is based on Oersted's experiment, but 
it covers much more. It contains the recognition that 
the association of an electric field, changing in time, 
with a magnetic field is essential for our further argu- 

As long as a charge is at rest there is only an electro- 


static field. But a magnetic field appears as soon as the 
charge begins to move. We can say more. The mag- 
netic field created by the motion of the charge will be 
stronger if the charge is greater and if it moves faster. 
This also is a consequence of Rowland's experiment. 
Once again using the field language, we can say: the 
faster the electric field changes, the stronger the ac- 
companying magnetic field. 

We have tried here to translate familiar facts from 
the language of fluids, constructed according to the 
old mechanical view, into the new language of fields. 
We shall see later how clear, instructive, and far- 
reaching our new language is. 


"The change of an electric field is accompanied by a 
magnetic field." If we interchange the words "mag- 
netic" and "electric", our sentence reads: "The change 
of a magnetic field is accompanied by an electric field." 
Only an experiment can decide whether or not this 
statement is true. But the idea of formulating this 
problem is suggested by the use of the field language. 

Just over a hundred years ago, Faraday performed 
an experiment which led to the great discovery of in- 
duced currents. 

The demonstration is very simple. We need only a 
solenoid or some other circuit, a bar magnet, and one 
of the many types of apparatus for detecting the exist- 
ence of an electric current. To begin with, a bar magnet 
is kept at rest near a solenoid which forms a closed 


circuit. No current flows through the wire, for no 
source is present. There is only the magnetostatic field 
of the bar magnet which does not change with time. 
Now, we quickly change the position of the magnet 
either by removing it or by bringing it nearer the 
solenoid, whichever we prefer. At this moment, a current 
will appear for a very short time and then vanish. 

Whenever the position of the magnet is changed, the 
current reappears, and can be detected by a sufficiently 
sensitive apparatus. But a current from the point of 
view of the field theory means the existence of an 
electric field forcing the flow of the electric fluids 
through the wire. The current, and therefore the elec- 
tric field, too, vanishes when the magnet is again at 

Imagine for a moment that the field language is 
unknown and the results of this experiment have to 
be described, qualitatively and quantitatively, in the 
language of old mechanical concepts. Our experiment 
then shows : by the motion of a magnetic dipole a new 
force was created, moving the electric fluid in the wire. 
The next question would be: upon what does this 
force depend? This would be very difficult to answer. 


We should have to investigate the dependence of the 
force upon the velocity of the magnet, upon its shape, 
and upon the shape of the circuit. Furthermore, this 
experiment, if interpreted in the old language, gives us 
no hint at all as to whether an induced current can be 
excited by the motion of another circuit carrying a 
current, instead of by motion of a bar magnet. 

It is quite a different matter if we use the field 
language and again trust our principle that the action is 
determined by the field. We see at once that a solenoid 
through which a current flows would serve as well as a 
bar magnet. The drawing shows two solenoids: one, 
small, through which a current flows, and the other, 
in which the induced current is detected, larger. We 

could move the small solenoid, as we previously moved 
the bar magnet, creating an induced current in the 
larger solenoid. Furthermore, instead of moving the 
small solenoid, we could create and destroy a magnetic 
field by creating and destroying the current, that is, 
by opening and closing the circuit. Once again, new 
facts suggested by the field theory are confirmed by 
experiment ! 

Let us take a simpler example. We have a closed wire 


without any source of current. Somewhere in the 
vicinity is a magnetic field. It means nothing to us 
whether the source of this magnetic field is another 
circuit through which an electric current flows, or a 

bar magnet. Our drawing shows the closed circuit and 
the magnetic lines of force. The qualitative and quanti- 
tative description of the induction phenomena is very 
simple in terms of the field language. As marked on 
the drawing, some lines of force go through the surface 
bounded by the wire. We have to consider the lines of 
force cutting that part of the plane which has the wire 
for a rim. No electric current is present so long as the 
field does not change, no matter how great its strength. 
But a current begins to flow through the rim-wire as 
soon as the number of lines passing through the surface 
surrounded by wire changes. The current is deter- 
mined by the change, however it may be caused, of the 
number of lines passing the surface. This change in the 
number of lines of force is the only essential concept 


for both the qualitative and the quantitative descrip- 
tions of the induced current. "The number of lines 
changes' 5 means that the density of the lines changes 
and this, we remember, means that the field strength 

These then are the essential points in our chain of 
reasoning: change of magnetic field -> induced cur- 
rent-emotion of charge -> existence of an electric 

Therefore : a changing magnetic field is accompanied by 
an electric field. 

Thus we have found the two most important pillars 
of support for the theory of the electric and magnetic 
field. The first is the connection between the changing 
electric field and the magnetic field. It arose from 
Oersted's experiment on the deflection of a magnetic 
needle and led to the conclusion: a changing electric field 
is accompanied by a magnetic field. 

The second connects the changing magnetic field 
with the induced current and arose from Faraday's 
experiment. Both formed a basis for quantitative de- 

Again the electric field accompanying the changing 
magnetic field appears as something real. We had to 
imagine, previously, the magnetic field of a current 
existing without the testing pole. Similarly, we must 
claim here that the electric field exists without the wire 
testing the presence of an induced current. 

In fact, our two-pillar structure could be reduced to 
only one, namely, to that based on Oersted's experi- 


ment. The result of Faraday's experiment could be 
deduced from this with the law of conservation of 
energy. We used the two-pillared structure only for the 
sake of clearness and economy. 

One more consequence of the field description should 
be mentioned. There is a circuit through which a cur- 
rent flows, with, for instance, a voltaic battery as the 
source of the current. The connection between the wire 
and the source of the current is suddenly broken. There 
is, of course, no current now ! But during this short 
interruption an intricate process takes place, a process 
which could again have been foreseen by the field 
theory. Before the interruption of the current, there 
was a magnetic field surrounding the wire. This ceased 
to exist the moment the current was interrupted. There- 
fore, through the interruption of a current, a magnetic 
field disappeared. The number of lines offeree passing 
through the surface surrounded by the wire changed 
very rapidly. But such a rapid change, however it 
is produced, must create an induced current. What 
really matters is the change of the magnetic field 
making the induced current stronger if the change is 
greater. This consequence is another test for the theory. 
The disconnection of a current must be accompanied 
by the appearance of a strong, momentary induced 
current. Experiment again confirms the prediction. 
Anyone who has ever disconnected a current must 
have noticed that a spark appears. This spark reveals 
the strong potential differences caused by the rapid 
change of the magnetic field. 


The same process can be looked at from a different 
point of view, that of energy. A magnetic field dis- 
appeared and a spark was created. A spark represents 
energy, therefore so also must the magnetic field. To 
use the field concept and its language consistently, we 
must regard the magnetic field as a store of energy. 
Only in this way shall we be able to describe the electric 
and magnetic phenomena in accordance with the law 
of conservation of energy. 

Starting as a helpful model, the field became more 
and more real. It helped us to understand old facts and 
led us to new ones. The attribution of energy to the 
field is one step farther in the development in which 
the field concept was stressed more and more, and the 
concepts of substances, so essential to the mechanical 
point of view, were more and more suppressed. 


The quantitative, mathematical description of the laws 
of the field is summed up in what are called Maxwell's 
equations. The facts mentioned so far led to the formu- 
lation of these equations, but their content is much 
richer than we have been able to indicate. Their simple 
form conceals a depth revealed only by careful study. 

The formulation of these equations is the most im- 
portant event in physics since Newton's time, not only 
because of their wealth of content, but also because 
they form a pattern for a new type of law. 

The characteristic features of Maxwell's equations, 
appearing in all other equations of modern physics, are 


summarized in one sentence. Maxwell's equations are 
laws representing the structure of the field. 

Why do Maxwell's equations differ in form and 
character from the equations of classical mechanics? 
What does it mean that these equations describe the 
structure of the field? How is it possible that, from the 
results of Oersted's and Faraday's experiments, we can 
form a new type of law, which proves so important for 
the further development of physics? 

We have already seen, from Oersted's experiment, 
how a magnetic field coils itself around a changing 
electric field. We have seen, from Faraday's experiment, 
how an electric field coils itself around a changing 
magnetic field. To outline some of the characteristic 
features of Maxwell's theory, let us, for the moment, 
focus all our attention on one of these experiments, 
say, on that of Faraday. We repeat the drawing in 

which an electric current is induced by a changing mag- 
netic field. We already know that an induced current 


appears if the number of lines of force, passing the sur- 
face bounded by the wire, changes. Then the current 
will appear if the magnetic field changes or the circuit 
is deformed or moved : if the number of magnetic lines 
passing through the surface is changed, no matter how 
this change is caused. To take into account all these 
various possibilities, to discuss their particular influ- 
ences, would necessarily lead to a very complicated 
theory. But can we not simplify our problem? Let us 
try to eliminate from our considerations everything 
which refers to the shape of the circuit, to its length, 
to the surface enclosed by the wire. Let us imagine 
that the circuit in our last drawing becomes smaller 
and smaller, shrinking gradually to a very small circuit 
enclosing a certain point in space. Then everything 
concerning shape and size is quite irrelevant. In this 
limiting process where the closed curve shrinks to a 
point, size and shape automatically vanish from our 
considerations and we obtain laws connecting changes 
of magnetic and electric field at an arbitrary point in 
space at an arbitrary instant. 

Thus, this is one of the principal steps leading to 
Maxwell's equations. It is again an idealized experiment 
performed in imagination by repeating Faraday's ex- 
periment with a circuit shrinking to a point. 

We should really call it half a step rather than a 
whole one. So far our attention has been focused on 
Faraday's experiment. But the other pillar of the field 
theory, based on Oersted's experiment, must be con- 
sidered just as carefully and in a similar manner. In this 


experiment the magnetic lines of force coil themselves 
around the current. By shrinking the circular magnetic 
lines of force to a point, the second half-step is per- 
formed and the whole step yields a connection between 
the changes of the magnetic and electric fields at an 
arbitrary point in spacfc and at an arbitrary instant. 

But still another essential step is necessary. According 
to Faraday's experiment, there must be a wire testing 
the existence of the electric field, just as there must be 
a magnetic pole, or needle, testing the existence of a 
magnetic field in Oersted's experiment. But Maxwell's 
new theoretical idea goes beyond these experimental 
facts. The electric and magnetic field or, in short, the 
electromagnetic field is, in Maxwell's theory, something 
real. The electric field is produced by a changing 
magnetic field, quite independently, whether or not 
there is a wire to test its existence; a magnetic field is 
produced by a changing electric field, whether or not 
there is a magnetic pole to test its existence. 

Thus two essential steps led to Maxwell's equations. 
The first: in considering Oersted's and Rowland's ex- 
periments, the circular line of the magnetic field coiling 
itself around the current and the changing electric field 
had to be shrunk to a point; in considering Faraday's 
experiment, the circular line of the electric field coiling 
itself around the changing magnetic field had to be 
shrunk to a point. The second step consists of the 
realization of the field as something real; the electro- 
magnetic field once created exists, acts, and changes 
according to Maxwell's laws. 


Maxwell's equations describe the structure of the 
electromagnetic field. All space is the scene of these 
laws and not, as for mechanical laws, only points in 
which matter or charges are present. 

We remember how it was in mechanics. By knowing 
the position and velocity of a particle at one single 
instant, by knowing the acting forces, the whole future 
path of the particle could be foreseen. In Maxwell's 
theory, if we know the field at one instant only, we 
can deduce from the equations of the theory how the 
whole field will change in space and time. Maxwell's 
equations enable us to follow the history of the field, 
just as the mechanical equations enabled us to follow 
the history of material particles. 

But there is still one essential difference between 
mechanical laws and Maxwell's laws. A comparison 
of Newton's gravitational laws and Maxwell's field 
laws will emphasize some of the characteristic features 
expressed by these equations. 

With the help of Newton's laws we can deduce the 
motion of the earth from the force acting between the 
sun and the earth. The laws connect the motion of the 
earth with the action of the far-off sun. The earth and 
the sun, though so far apart, are both actors in the play 
of forces. 

In Maxwell's theory there are no material actors. 
The mathematical equations of this theory express the 
laws governing the electromagnetic field. They do not, 
as in Newton's laws, connect two widely separated 
events; they do not connect the happenings here with 


the conditions there. The field here and now depends 
on the field in the immediate neighbourhood at a time 
just past. The equations allow us to predict what will 
happen a little farther in space and a little later in time, 
if we know what happens here and now. They allow 
us to increase our knowledge of the field by small steps. 
We can deduce what happens here from that which 
happened far away by the summation of these very 
small steps. In Newton's theory, on the contrary, only 
big steps connecting distant events are permissible. The 
experiments of Oersted and Faraday can be regained 
from Maxwell's theory, but only by the summation of 
small steps each of which is governed by Maxwell's 

A more thorough mathematical study of Maxwell's 
equations shows that new and really unexpected con- 
clusions can be drawn and the whole theory submitted 
to a test on a much higher level, because the theoretical 
consequences are now of a quantitative character and 
are revealed by a whole chain of logical arguments. 

Let us again imagine an idealized experiment. A small 
sphere with an electric charge is forced, by some ex- 
ternal influence, to oscillate rapidly and in a rhythmical 
way, like a pendulum. With the knowledge we already 
have of the changes of the field, how shall we describe 
everything that is going on here, in the field language? 

The oscillation of the charge produces a changing 
electric field. This is always accompanied by a chang- 
ing magnetic field. If a wire forming a closed circuit 
is placed in the vicinity, then again the changing 


magnetic field will be accompanied by an electric current 
in the circuit. All this is merely a repetition of known 
facts, but the study of Maxwell's equations gives a 
much deeper insight into the problem of the oscillating 
electric charge. By mathematical deduction from Max- 
well's equations we can detect the character of the 
field surrounding an oscillating charge, its structure 
near and far from the source and its change with time. 
The outcome of such deduction is the electromagnetic 
wave. Energy radiates from the oscillating charge 
travelling with a definite speed through space; but 
a transference of energy, the motion of a state, is 
characteristic of all wave phenomena. 

Different types of waves have already been consid- 
ered. There was the longitudinal wave caused by the 
pulsating sphere, where the changes of density were 
propagated through the medium. There was the jelly- 
like medium in which the transverse wave spread. A 
deformation of the jelly, caused by the rotation of the 
sphere, moved through the medium. What kind of 
changes are now spreading in the case of an electro- 
magnetic wave? Just the changes of an electromagnetic 
field ! Every change of an electric field produces a 
magnetic field; every change of this magnetic field 
produces an electric field; every change of..., and so 
on. As field represents energy, all these changes spread- 
ing out in space, with a definite velocity, produce a 
wave. The electric and magnetic lines of force always 
lie, as deduced from the theory, on planes perpen- 
dicular to the direction of propagation. The wave 


produced is, therefore, transverse. The original features 
of the picture of the field we formed from Oersted's 
and Faraday's experiments are still preserved, but we 
now recognize that it has a deeper meaning. 

The electromagnetic wave spreads in empty space. 
This, again, is a consequence of the theory. If the os- 
cillating charge suddenly ceases to move, then its field 
becomes electrostatic. But the series of waves created 
by the oscillation continues to spread. The waves lead 
an independent existence and the history of their 
changes can be followed just as that of any other 
material object. 

We understand that our picture of an electromag- 
netic wave, spreading with a certain velocity in space 
and changing in time, follows from Maxwell's equa- 
tions only because they describe the structure of the 
electromagnetic field at any point in space and for any 

There is another very important question. With what 
speed does the electromagnetic wave spread in empty 
space? The theory, with the support of some data from 
simple experiments having nothing to do with the 
actual propagation of waves, gives a clear answer: the^ 
velocity of an electromagnetic wave is equal to the velocity qf 

lijjrz ~ 

Oersted's and Faraday's experiments formed the 
basis on which Maxwell's laws were built. All our 
results so far have come from a careful study of these 
laws, expressed in the field language. The theoretical 
discovery of an electromagnetic wave spreading with 


the speed of light is one of the greatest achievements in 
the history of science. 

Experiment has confirmed the prediction of theory. 
Fifty years ago, Hertz proved, for the first time, the 
existence of electromagnetic waves and confirmed ex- 
perimentally that their velocity is equal to that of 
light. Nowadays, millions of people demonstrate that 
electromagnetic waves are sent and received. Their 
apparatus is far more complicated than that used by 
Hertz and detects the presence of waves thousands of 
miles from their sources instead of only a few yards. 


The electromagnetic wave is a transverse one and is 
propagated with the velocity of light in empty space. 
The fact that their velocities are the same suggests a 
close relationship between optical and electromagnetic 

When we had to choose between the corpuscular 
and the wave theory, we decided in favour of the 
wave theory. The diffraction of light was the strongest 
argument influencing our decision. But we shall not 
contradict any of the explanations of the optical facts 
by also assuming that the light wave is an electromagnetic 
one. On the contrary, still other conclusions can be 
drawn. If this is really so, then there must exist some 
connection between the optical and electrical properties 
of matter that can be deduced from the theory. The 
fact that conclusions of this kind can really be drawn 


and that they stand the test of experiment is an essential 
argument in favour of the electromagnetic theory of 

This great result is due to the field theory. Two 
apparently unrelated branches of science are covered 
by the same theory. The same Maxwell's equations 
describe both electric induction and optical refraction. If 
it is our aim to describe everything that ever happened 
or may happen with the help of one theory, then the 
union of optics and electricity is, undoubtedly, a very 
great step forward. From the physical point of view, 
the only difference between an ordinary electromag- 
netic wave and a light wave is the wave-length: this is 
very small for light waves, detected by the human eye, 
and great for ordinary electromagnetic waves, detected 
by a radio receiver. 

The old mechanical view attempted to reduce all 
events in nature to forces acting between material par- 
ticles. Upon this mechanical view was based the first 
naive theory of the electric fluids. The field did not 
exist for the physicist of the early years of the nine- 
teenth century. For him only substance and its changes 
were real. He tried to describe the action of two elec- 
tric charges only by concepts referring directly to the 
two charges. 

In the beginning, the field concept was no more than 
a means of facilitating the understanding of phenomena 
from the mechanical point of view. In the new field 
language it is the description of the field between the 
two charges, and not the charges themselves, which is 


essential for an understanding of their action. The recog- 
nition of the new concepts grew steadily, until substance 
was overshadowed by the field. It was realized that 
something of great importance had happened in physics. 
A new reality was created, a new concept for which 
there was no place in the mechanical description. 
Slowly and by a struggle the field concept established 
for itself a leading place in physics and has remained 
one of the basic physical concepts. The electromagnetic 
field is, for the modern physicist, as real as the chair on 
which he sits. 

But it would be unjust to consider that the new field 
view freed science from the errors of the old theory of 
electric fluids or that the new theory destroys the 
achievements of the old. The new theory shows the 
merits as well as the limitations of the old theory and 
allows us to regain our old concepts from a higher 
level. This is true not only for the theories of electric 
fluids and field, but for all changes in physical theories, 
however revolutionary they may seem. In our case, 
we still find, for example, the concept of the electric 
charge in Maxwell's theory, though the charge is un- 
derstood only as a source of the electric field. Cou- 
lomb's law is still valid and is contained in Maxwell's 
equations from which it can be deduced as one of the 
many consequences. We can still apply the old theory, 
whenever facts within the region of its validity are in- 
vestigated. But we may as well apply the new theory, 
since all the known facts are contained in the realm of 
its validity. 


To use a comparison, we could say that creating a 
new theory is not like destroying an old barn and erect- 
uTits piarq, ^lt is rather like climEing 

a mountain, gaining new and wider views, discovering 
cuimeaions b^twe^Trouf^IarnHg^point and 

its rich environment. But the point from which we 
started out still exists and can be seen, although it 
appears smaller and forms a tiny part of our broad 
view gained by the mastery of the obstacles on our 
adventurous way up. 

It was, indeed, a long time before the full content of 
Maxwell's theory was recognized. The field was at first 
considered as something which might later be inter- 
preted mechanically with the help of ether. By the 
time it was realized that this programme could not be 
carried out, the achievements of the field theory had 
already become too striking and important for it to be 
exchanged for a mechanical dogma. On the other hand, 
the problem of devising the mechanical model of ether 
seemed to become less and less interesting and the result, 
in view of the forced and artificial character of the 
assumptions, more and more discouraging. 

Our only way out seems to be to take for granted 
the fact that space has the physical property of trans- 
mitting electromagnetic waves, and not to bother too 
much about the meaning of this statement. We may 
still use the word ether, but only to express some phys- 
ical property of space. This word ether has changed its 
meaning many times in the development of science. At 
the moment it no longer stands for a medium built up 


of particles. Its story, by no means finished, is continued 
by the relativity theory. 


On reaching this stage of our story, we must turn back 
to the beginning, to Galileo's law of inertia. We quote 
once more : 

Every body perseveres in its state of rest, or of uniform 
motion in a right line, unless it is compelled to change that 
state by forces impressed thereon. 

Once the idea of inertia is understood, one wonders 
what more can be said about it. Although this problem 
has already been thoroughly discussed, it is by no means 

Imagine a serious scientist who believes that the law 
of inertia can be proved or disproved by actual experi- 
ments. He pushes small spheres along a horizontal 
table, trying to eliminate friction so far as possible. He 
notices that the motion becomes more uniform as the 
table and the spheres are made smoother. Just as he is 
about to proclaim the principle of inertia, someone 
suddenly plays a practical joke on him. Our physicist 
works in a room without windows and has no com- 
munication whatever with the outside world. The 
practical joker installs some mechanism which enables 
him to cause the entire room to rotate quickly on an 
axis passing through its centre. As soon as the rotation 
begins, the physicist has new and unexpected experi- 
ences. The sphere which has been moving uniformly 

tries to get as far away from the centre and as near to 
the walls of the room as possible. He himself feels a 
strange force pushing him against the wall. He ex- 
periences the same sensation as anyone in a train or 
car travelling fast round a curve, or even more, on a 
rotating merry-go-round. All his previous results go to 

Our physicist would have to discard, with the law 
of inertia, all mechanical laws. The law of inertia was 
his starting-point; if this is changed, so are all his further 
conclusions. An observer destined to spend his whole 
life in the rotating room, and to perform all his experi- 
ments there, would have laws of mechanics differing 
from ours. If, on the other hand, he enters the room 
with a profound knowledge and a firm belief in the 
principles of physics, his explanation for the apparent 
breakdown of mechanics would be the assumption that 
the room rotates. By mechanical experiments he could 
even ascertain how it rotates. 

Why should we take so much interest in the ob- 
server in his rotating room? Simply because we, on 
our earth, are to a certain extent in the same position. 
Since the time of Copernicus we have known that the 
earth rotates on its axis and moves around the sun. 
Even this simple idea, so clear to everyone, was not left 
untouched by the advance of science. But let us leave 
this question for the time being and accept Copernicus' 
point of view. If our rotating observer could not con- 
firm the laws of mechanics, w^- on our earth, should 
also be unable to do so. But the rotation of the earth is 


comparatively slow, so that the effect is not very dis- 
tinct. Nevertheless, there are many experiments which 
show a small deviation from the mechanical laws, and 
their consistency can be regarded as proof of the rota- 
tion of the earth. 

Unfortunately we cannot place ourselves between 
the sun and the earth, to prove there the exact validity 
of the law of inertia and to get a view of the rotating 
earth. This can be done only in imagination. All our 
experiments must be performed on the earth on which 
we are compelled to live. The same fact is often ex- 
pressed more scientifically: the earth is our co-ordinate 

To show the meaning of these words more clearly, 
let us take a simple example. We can predict the posi- 
tion, at any time, of a stone thrown from a tower, and 
confirm our prediction by observation. If a measuring- 
rod is placed beside the tower, we can foretell with 
what mark on the rod the falling body will coincide 
at any given moment. The tower and scale must, ob- 
viously, not be made of rubber or any other material 
which would undergo any change during the experi- 
ment. In fact, the unchangeable scale, rigidly con- 
nected with the earth, and a good clock are all we 
need, in principle, for the experiment. If we have these, 
we can ignore not only the architecture of the tower, 
but its very presence. The foregoing assumptions are 
all trivial and not usually specified in descriptions of 
such experiments. But this analysis shows how many 
hidden assumptions there are in every one of our state- 


ments. In our case, we assumed the existence of a rigid 
bar and an ideal clock, without which it would be im- 
possible to check Galileo's law for falling bodies. With 
this simple but fundamental physical apparatus, a rod 
and a clock, we can confirm this mechanical law with 
a certain degree of accuracy. Carefully performed, this 
experiment reveals discrepancies between theory and 
experiment due to the rotation of the earth or, in other 
words, to the fact that the laws of mechanics, as here 
formulated, are not strictly valid in a co-ordinate sys- 
tem rigidly connected with the earth. 

In all mechanical experiments, no matter of what 
type, we have to determine positions of material points 
at some definite time, just as in the above experiment 
with a falling body. But the position must always be 
described with respect to something, as in the previous 
case to the tower and the scale. We must have what we 
call some frame of reference, a mechanical scaffold, to 
be able to determine the positions of bodies. In de- 
scribing the positions of objects and men in a city, the 
streets and avenues form the frame to which we refer. 
So far we have not bothered to describe the frame 
when quoting the laws of mechanics, because we 
happen to live on the earth and there is no difficulty 
in any particular case in fixing a frame of reference, 
rigidly connected with the earth. This frame, to which 
we refer all our observations, constructed of rigid un- 
changeable bodies, is called the co-ordinate system. Since 
this expression will be used very often, we shall simply 
write c.s. 



All our physical statements thus far have lacked 
something. We took no notice of the fact that all ob- 
servations must be made in a certain c.s. Instead of 
describing the structure of this c.s., we just ignored its 
existence. For example, when we wrote " a body moves 
uniformly. . ." we should really have written, "a body 

moves uniformly, relative to a chosen c.s " Our 

experience with the rotating room taught us that the 
results of mechanical experiments may depend on the 
c.s. chosen. 

If two c.s. rotate with respect to each other, then the 
laws of mechanics cannot be valid in both. If the sur- 
face of the water in a swimming pool, forming one of 
the co-ordinate systems, is horizontal, then in the other 
the surface of the water in a similar swimming pool 
takes the curved form similar to anyone who stirs his 
coffee with a spoon. 

When formulating the principal clues of mechanics 
we omitted one important point. We did not state^for 
which c.s. they arej^aiuL- JborlHis reason, the whole of 

lid-air since we do not 

know to which frame it refers. Let us, however, pass 
over mis ditliculty for the moment. We shall make the 
slightly incorrect assumption that in every c.s. rigidly 
connected with the earth the laws of classical me- 
chanics are valid. This is done in order to fix the c.s. 
and to make our statements definite. Although our 
statement that the earth is a suitable frame of reference 
is not wholly correct, we shall accept it for the present. 
We assume, therefore, the existence of one c.s. for 


which the laws of mechanics are valid. Is this the only 
one? Suppose we have a c.s. such as a train, a ship or 
an aeroplane moving relative to our earth. Will the 
laws of mechanics be valid for these new c.s.? We 
know definitely that they are not always valid, as for 
instance in the case of a train turning a curve, a ship 
tossed in a storm or an aeroplane in a tail spin. Let us 
begin with a simple example. A c.s. moves uniformly, 
relative to our "good" c.s., that is, one in which the 
laws of mechanics are valid. For instance, an ideal 
train or a ship sailing with delightful smoothness along 
a straight line and with a never-changing speed. We 
know from everyday experience that both systems will 
be "good", that physical experiments performed in a 
uniformly moving train or ship will give exactly the 
same results as on the earth. But, if the train stops, or 
accelerates abruptly, or if the sea is rough, strange 
things happen. In the train, the trunks fall off the 
luggage racks; on the ship, tables and chairs are thrown 
about and the passengers become seasick. From the 
physical point of view this simply means that the laws 
of mechanics cannot be applied to these c.s., that they 
are " bad" c.s. 

This result can be expressed by the so-called Galilean 
relativity principle : if the laws of mechanics are valid in one 
c.s., then they are valid in any other c.s. moving uniformly 
relative to the first. 

If we have two c.s. moving non-uniformly, relative 
to each other, then the laws of mechanics cannot be 
valid in both. "Good" co-ordinate systems, that is, 


those for which the laws of mechanics are valid, we 
call inertial systems. The question as to whether an in- 
ertial system exists at all is still unsettled. But if there 
is one such system, then there is an infinite number of 
them. Every c.s. moving uniformly, relative to the 
initial one, is also an inertial c.s. 

Let us consider the case of two c.s. starting from a 
known position and moving uniformly, one relative 
to the other, with a known velocity. One who prefers 
concrete pictures can safely think of a ship or a train 
moving relative to the earth. The laws of mechanics 
can be confirmed experimentally with the same degree 
of accuracy, on the earth or in a train or on a ship 
moving uniformly. But some difficulty arises if the 
observers of two systems begin to discuss observations 
of the same event from the point of view of their 
different c.s. Each would like to translate the other's 
observations into his own language. Again a simple 
example: the same motion of a particle is observed 
from two c.s. the earth and a train moving uniformly. 
These are both inertial. Is it sufficient to know what is 
observed in one c.s. in order to find out what is ob- 
served in the other, if the relative velocities and positions 
of the two c.s. at some moment are known? It is most 
essential, for a description of events, to know how to 
pass from one c.s. to another, since both c.s. are equiva- 
lent and both equally suited for the description of events 
in nature. Indeed, it is quite enough to know the 
results obtained by an observer in one c.s. to know 
those obtained by an observer in the other. 


Let us consider the problem more abstractly, without 
ship or train. To simplify matters we shall investigate 
only motion along straight lines. We have, then, a 
rigid bar with a scale and a good clock. The rigid 
bar represents, in the simple case of rectilinear motion, 
a c.s. just as did the scale on the tower in Galileo's 
experiment. It is always simpler and better to think of 
a c.s. as a rigid bar in the case of rectilinear motion and 
a rigid scaffold built of parallel and perpendicular rods 
in the case of arbitrary motion in space, disregarding 
towers, walls, streets, and the like. Suppose we have, in 
our simplest case, two c.s., that is, two rigid rods; we 
draw one above the other and call them respectively 
the "upper" and "lower" c.s. We assume that the two 
c.s. move with a definite velocity relative to each other, 
so that one slides along the other. It is safe to assume 
that both rods are of infinite length and have initial 
points but no end-points. One clock is sufficient for 
the two c.s., for the time flow is the same for both. 
When we begin our observation the starting-points of 
the two rods coincide. The position of a material point 
is characterized, at this moment, by the same number in 
both c.s. The material point coincides with a point on 
the scale on the rod, thus giving us a number deter- 
mining the position of this material point. But, if the 
rods move uniformly, relative to each other, the numbers 
corresponding to the positions will be different after 
some time, say, one second. Consider a material point 
resting on the upper rod. The number determining its 
position on the upper c.s. does not change with time. 


But the corresponding number for the lower rod will 
change. Instead of "the number corresponding to a 
position of the point", we shall say briefly, the co- 
ordinate of a point. Thus we see from our drawing that 

< >E 

although the following sentence sounds intricate, it is 
nevertheless correct and expresses something very simple. 
The co-ordinate of a point in the lower c.s. is equal to 
its co-ordinate in the upper c.s. plus the co-ordinate of 
the origin of the upper c.s. relative to the lower c.s. 
The important thing is that we can always calculate 
the position of a particle in one c.s. if we know the 
position in the other. For this purpose we have to 
know the relative positions of the two co-ordinate sys- 
tems in question at every moment. Although all this 
sounds learned, it is, really, very simple and hardly 
worth such detailed discussion, except that we shall 
find it useful later. 

It is worth our while to notice the difference be- 
tween determining the position of a point and the time 
of an event. Every observer has his own rod which 
forms his c.s., but there is only one clock for them all, 
Time is something " absolute" which flows in the same 
way for all observers in all c.s. 

Now another example. A man strolls with a velocity 
of three miles per hour along the deck of a large ship. 


This is his velocity relative to the ship, or, in other 
words, relative to a c.s. rigidly connected with the ship. 
If the velocity of the ship is thirty miles per hour relative 
to the shore, and if the uniform velocities of man and 
ship both have the same direction, then the velocity 
of the stroller will be thirty-three miles per hour relative 
to an observer on the shore, or three miles per hour 
relative to the ship. We can formulate this fact more 
abstractly: the velocity of a moving material point, 
relative to the lower c.s., is equal to that relative to 
the upper c.s. plus or minus the velocity of the upper 
c.s. relative to the lower, depending upon whether the 
velocities have the same or opposite directions. We 
can, therefore, always transform not only positions, 

but also velocities from one c.s. to another if we know 
the realtive velocities of the two c.s. The positions, or 
co-ordinates, and velocities are examples of quantities 
which are different in different c.s. bound together by 
certain, in this case very simple, transformation laws. 

There exist quantities, however, which are the same 
in both c.s. and for which no transformation laws are 
needed. Take as an example not one, but two fixed 
points on the upper rod and consider the distance 
between them. This distance is the difference in the 
co-ordinates of the two points. To find the positions 


of two points relative to different c.s., we have to use 
transformation laws. But in constructing the differ- 
ences of two positions the contributions due to the 

different c.s. cancel each other and disappear, as is 
evident from the drawing. We have to add and sub- 
tract the distance between the origins of two c.s. The 
distance of two points is, therefore, invariant^ that is, 
independent of the choice of the c.s. 

The next example of a quantity independent of the 
c.s. is the change of velocity, a concept familiar to us 
from mechanics. Again, a material point moving along 
a straight line is observed from two c.s. Its change of 
velocity is, for the observer in each c.s., a difference 
between two velocities, and the contribution due to 
the uniform relative motion of the two c.s. disappears 
when the difference is calculated. Therefore, the change 
of velocity is an invariant, though only, of course, on 
condition that the relative motion of our two c.s. is 
uniform. Otherwise, the change of velocity would be 
different in each of the two c.s., the difference being 
brought in by the change of velocity of the relative 
motion of the two rods, representing our co-ordinate 

Now the last example ! We have two material points, 
with forces acting between them which depend only 


on the distance. In the case of rectilinear motion, the 
distance, and therefore the force as well, is invariant. 
Newton's law connecting the force with the change of 
velocity will, therefore, be valid in both c.s. Once 
again we reach a conclusion which is confirmed by 
everyday experience : if the laws of mechanics are valid 
in one c.s., then they are valid in all c.s. moving uni- 
formly with respect to that one. Our example was, of 
course, a very simple one, that of rectilinear motion in 
which the c.s. can be represented by a rigid rod. But 
our conclusions are generally valid, and can be sum- 
marized as follows : 

(1) We know of no rule for finding an inertial system. 
Given one, however, we can find an infinite num- 
ber, since all c.s. moving uniformly, relative to each 
other, are inertial systems if one of them is. 

(2) The time corresponding to an event is the same in 
all c.s. But the co-ordinates and velocities are dif- 
ferent, and change according to the transformation 

(3) Although the co-ordinates and velocity change 
when passing from one c.s. to another, the force 
and change of velocity, and therefore the laws of 
mechanics are invariant with respect to the trans- 
formation laws. 

The laws of transformation formulated here for co- 
ordinates and velocities we shall call the transforma- 
tion laws of classical mechanics, or more briefly, the 
classical transformation. 



The Galilean relativity principle is valid for mechanical 
phenomena. The same laws of mechanics apply to all 
inertial systems moving relative to each other. Is this 
principle also valid for non-mechanical phenomena, 
especially for those for which the field concepts proved 
so very important? All problems concentrated around 
this question immediately bring us to the starting-point 
of the relativity theory. 

We remember that the velocity of light in vacuo, or 
in other words, in ether, is 186,000 miles per second 
and that light is an electromagnetic wave spreading 
through the ether. The electromagnetic field carries 
energy which, once emitted from its source, leads an 
independent existence. For the time being, we shall 
continue to believe that the ether is a medium through 
which electromagnetic waves, and thus also light waves, 
are propagated, even though we are fully aware of the 
many difficulties connected with its mechanical structure. 

We are sitting in a closed room so isolated from the 
external world that no air can enter or escape. If we sit 
still and talk we are, from the physical point of view, 
creating sound waves, which spread from their resting 
source with the velocity of sound in air. If there were 
no air or other material medium between the mouth 
and the ear, we could not detect a sound. Experiment 
has shown that the velocity of sound in air is the same 
in all directions, if there is no wind and the air is at 
rest in the chosen c.s. 


Let us now imagine that our room moves uniformly 
through space. A man outside sees, through the glass 
walls of the moving room (or train if you prefer), every- 
thing which is going on inside. From the measure- 
ments of the inside observer he can deduce the velocity 
of sound relative to his c.s. connected with his sur- 
roundings, relative to which the room moves. Here 
again is the old, much discussed, problem of deter- 
mining the velocity in one c.s. if it is already known 
in another. 

The observer in the room claims: the velocity of 
sound is, for me, the same in all directions. 

The outside observer claims : the velocity of sound, 
spreading in the moving room and determined in my 
c.s., is not the same in all directions. It is greater than 
the standard velocity of sound in the direction of the 
motion of the room and smaller in the opposite direction. 

These conclusions are drawn from the classical trans- 
formation and can be confirmed by experiment. The 
room carries within it the material medium, the air 
through which sound waves are propagated, and the 
velocities of sound will, therefore, be different for the 
inside and outside observer. 

We can draw some further conclusions from the 
theory of sound as a wave propagated through a ma- 
terial medium. One way, though by no means the 
simplest, of not hearing what someone is saying, is to 
run, with a velocity greater than that of sound, rela- 
tive to the air surrounding the speaker. The sound 
waves produced will then never be able to reach our 


ears. On the other hand, if we missed an important 
word which will never be repeated, we must run with 
a speed greater than that of sound to reach the pro- 
duced wave and to catch the word. There is nothing 
irrational in either of these examples except that in 
both cases we should have to run with a speed of about 
four hundred yards per second, and we can very well 
imagine that further technical development will make 
such speeds possible. A bullet fired from a gun actually 
moves with a speed greater than that of sound and a 
man placed on such a bullet would never hear the 
sound of the shot. 

All these examples are of a purely mechanical charac- 
ter and we can now formulate the important questions : 
could we repeat what has just been said of a sound 
wave, in the case of a light wave? Do the Galilean 
relativity principle and the classical transformation 
apply to optical and electrical phenomena as well as to 
mechanical? It would be risky to answer these questions 
by a simple "yes" or "no" without going more deeply 
into their meaning. 

In the case of the sound wave in the room moving 
uniformly, relative to the outside observer, the follow- 
ing intermediate steps are very essential for our con- 
clusion : 
The moving room carries the air in which the sound 

wave is propagated. 
The velocities observed in two c.s. moving uniformly, 

relative to each other, are connected by the classical 



The corresponding problem for light must be for- 
mulated a little differently. The observers in the room 
are no longer talking, but are sending light signals, or 
light waves in every direction. Let us further assume 
that the sources emitting the light signals are perma- 
nently resting in the room. The light waves move 
through the ether just as the sound waves moved 
through the air. 

Is the ether carried with the room as the air was? 
Since we have no mechanical picture of the ether, it is 
extremely difficult to answer this question. If the room 
is closed, the air inside is forced to move with it. There 
is obviously no sense in thinking of ether in this way, 
since all matter is immersed in it and it penetrates 
everywhere. No doors are closed to ether. The "mov- 
ing room" now means only a moving c.s. to which the 
source of light is rigidly connected. It is, however, not 
beyond us to imagine that the room moving with its 
light source carries the ether along with it just as the 
sound source and air were carried along in the closed 
room. But we can equally well imagine the opposite: 
that the room travels through the ether as a ship 
through a perfectly smooth sea, not carrying any part 
of the medium along but moving through it. In our 
first picture, the room moving with its light source 
carries the ether. An analogy with a sound wave is 
possible and quite similar conclusions can be drawn. 
In the second, the room moving with its light source 
does not carry the ether. No analogy with a sound 
wave is possible and the conclusions drawn in the case 


of a sound wave do not hold for a light wave. These 
are the two limiting possibilities. We could imagine the 
still more complicated possibility that the ether is only 
partially carried by the room moving with its light 
source. But there is no reason to discuss the more 
complicated assumptions before finding out which of 
the two simpler limiting cases experiment favours. 

We shall begin with our first picture and assume, for 
the present: the ether is carried along by the room 
moving with its rigidly connected light source. If we 
believe in the simple transformation principle for the 
velocities of sound waves, we can now apply our con- 
clusions to light waves as well. There is no reason for 
doubting the simple mechanical transformation law 
which only states that the velocities have to be added 
in certain cases and subtracted in others. For the mo- 
ment, therefore, we shall assume both the carrying of 
the ether by the room moving with its light source and 
the classical transformation. 

If I turn on the light and its source is rigidly con- 
nected with my room, then the velocity of the light 
signal has the well-known experimental value 186,000 
miles per second. But the outside observer will notice 
the motion of the room, and, therefore, that of the 
source and, since the ether is carried along, his con- 
clusion must be: the velocity of light in my outside c.s. 
is different in different directions. It is greater than the 
standard velocity of light in the direction of the mo- 
tion of the room and smaller in the opposite direction. 
Our conclusion is: if ether is carried with the room 


moving with its light source and if the mechanical laws 
are valid, then the velocity of light must depend on 
the velocity of the light source. Light reaching our 
eyes from a moving light source would have a greater 
velocity if the motion is toward us and smaller if it is 
away from us. 

If our speed were greater than that of light, we 
should be able to run away from a light signal. We 
could see occurrences from the past by reaching pre- 
viously sent light waves. We should catch them in a 
reverse order to that in which they were sent, and the 
train of happenings on our earth would appear like a 
film shown backward, beginning with a happy ending. 
These conclusions all follow from the assumption that 
the moving c.s. carries along the ether and the me- 
chanical transformation laws are valid. If this is so, 
the analogy between light and sound is perfect. 

But there is no indication as to the truth of these 
conclusions. On the contrary, they are contradicted by 
all observations made with the intention of proving 
them. There is not the slightest doubt as to the clarity 
of this verdict, although it is obtained through rather 
indirect experiments in view of the great technical diffi- 
culties caused by the enormous value of the velocity of 
light. The velocity of light is always the same in all c.s. 
independent of whether or not the emitting source moves , or how 
it moves. 

We shall not go into detailed description of the many 
experiments from which this important conclusion can 
be drawn. We can, however, use some very simple 


arguments which, though they do not prove that the 
velocity of light is independent of the motion of the 
source, nevertheless make this fact convincing and un- 

In our planetary system the earth and other planets 
move around the sun. We do not know of the existence 
of other planetary systems similar to ours. There are, 
however, very many double-star systems, consisting of 
two stars moving around a point, called their centre of 
gravity. Observation of the motion of these double 
stars reveals the validity of Newton's gravitational law. 
Now suppose that the speed of light depends on the 
velocity of the emitting body. Then the message, that 
is, the light ray from the star, will travel more quickly 
or more slowly, according to the velocity of the star at 
the moment the ray is emitted. In this case the whole 
motion would be muddled and it would be impossible 
to confirm, in the case of distant double stars, the 
validity of the same gravitational law which rules over 
our planetary system. 

Let us consider another experiment based upon a 
very simple idea. Imagine a wheel rotating very quickly. 
According to our assumption, the ether is carried by 
the motion and takes a part in it. A light wave passing 
near the wheel would have a different speed when 
the wheel is at rest than when it is in motion. The 
velocity of light in ether at rest should differ from 
that in ether which is being quickly dragged round by 
the motion of the wheel, just as the velocity of a sound 
wave varies on calm and windy days. But no such 


difference is detected! No matter from which angle 
we approach the subject, or what crucial experiment 
we may devise, the verdict is always against the assump- 
tion of the ether carried by motion. Thus, the result 
of our considerations, supported by more detailed and 
technical argument, is : 
The velocity of light does not depend on the motion 

of the emitting source. 
It must not be assumed that the moving body carries 

the surrounding ether along. 

We must, therefore, give up the analogy between 
sound and light waves and turn to the second possi- 
bility: that all matter moves through the ether, which 
takes no part whatever in the motion. This means that 
we assume the existence of a sea of ether with all c.s. 
resting in it, or moving relative to it. Suppose we 
leave, for a while, the question as to whether experi- 
ment proved or disproved this theory. It will be better 
to become more familiar with the meaning of this new 
assumption and with the conclusions which can be 
drawn from it. 

There exists a c.s. resting relative to the ether-sea. 
In mechanics, not one of the many c.s. moving uni- 
formly, relative to each other, could be distinguished. 
All such c.s. were equally "good" or "bad". If we have 
two c.s. moving uniformly, relative to each other, it is 
meaningless, in mechanics, to ask which of them is in 
motion and which at rest. Only relative uniform mo- 
tion can be observed. We cannot talk about absolute 
uniform motion because of the Galilean relativity 


principle. What is meant by the statement that absolute 
and not only relative uniform motion exists? Simply 
that there exists one c.s. in which some of the laws of 
nature are different from those in all others. Also that 
every observer can detect whether his c.s. is at rest or 
in motion by comparing the laws valid in it with those 
valid in the only one which has the absolute monopoly 
of serving as the standard c.s. Here is a different state 
of affairs from classical mechanics, where absolute uni- 
form motion is quite meaningless because of Galileo's 
law of inertia. 

What conclusions can be drawn in the domain of 
field phenomena if motion through ether is assumed? 
This would mean that there exists one c.s. distinct from 
all others, at rest relative to the ether-sea. It is quite 
clear that some of the laws of nature must be different 
in this c.s., otherwise the phrase "motion through 
ether" would be meaningless. If the Galilean relativity 
principle is valid, then motion through ether makes no 
sense at all. It is impossible to reconcile these two ideas. 
If, however, there exists one special c.s. fixed by the 
ether, then to speak of "absolute motion" or "absolute 
rest" has a definite meaning. 

We really have no choice. We tried to save the 
Galilean relativity principle by assuming that systems 
carry the ether along in their motion, but this led to a 
contradiction with experiment. The only way out is to 
abandon the Galilean relativity principle and try out 
the assumption that all bodies move through the calm 


The next step is to consider some conclusions con- 
tradicting the Galilean relativity principle and sup- 
porting the view of motion through ether, and to put 
them to the test of an experiment. Such experiments 
are easy enough to imagine, but very difficult to per- 
form. As we are concerned here only with ideas, we 
need not bother with technical difficulties. 

Again we return to our moving room with two 
observers, one inside and one outside. The outside 
observer will represent the standard c.s., designated by 
the ether-sea. It is the distinguished c.s. in which the 
velocity of light always has the same standard value. 
All light sources, whether moving or at rest in the 
calm ether-sea, propagate light with the same velocity. 
The room and its observer move through the ether. 
Imagine that a light in the centre of the room is flashed 
on and off and, furthermore, that the walls of the room 
are transparent so that the observers, both inside and 
outside, can measure the velocity of the light. If we 
ask the two observers what results they expect to ob- 
tain, their answers would run something like this : 

The outside observer: My c.s. is designated by the 
ether-sea. Light in my c.s. always has the standard 
value. I need not care whether or not the source of 
light or other bodies are moving, for they never carry 
my ether-sea with them. My c.s. is distinguished from 
all others and the velocity of light must have its standard 
value in this c.s., independent of the direction of the 
light beam or the motion of its source. 

The inside observer: My room moves through the 


ether-sea. One of the walls runs away from the light 
and the other approaches it. If my room travelled, 
relative to the ether-sea, with the velocity of light, then 
the light emitted from the centre of the room would 
never reach the wall running away with the velocity 
of light. If the room travelled with a velocity smaller 
than that of light, then a wave sent from the centre of 
the room would reach one of the walls before the 
other. The wall moving toward the light wave would 
be reached before the one retreating from the light 
wave. Therefore, although the source of light is rigidly 
connected with my G.S., the velocity of light will not 
be the same in all directions. It will be smaller in the 
direction of the motion relative to the ether-sea as the 
wall runs away, and greater in the opposite direction 
as the wall moves toward the wave and tries to meet 
it sooner. 

Thus, only in the one c.s. distinguished by the ether- 
sea should the velocity of light be equal in all directions. 
For other c.s. moving relatively to the ether-sea it 
should depend on the direction in which we are 

The crucial experiment just considered enables us 
to test the theory of motion through the ether-sea. 
Nature, in fact, places at our disposal a system moving 
with a fairly high velocity the earth in its yearly mo- 
tion around the sun. If our assumption is correct, then 
the velocity of light in the direction of the motion of 
the earth should differ from the velocity of light in an 
opposite direction. The differences can be calculated 


and a suitable experimental test devised. In view of 
the small time-differences following from the theory, 
very ingenious experimental arrangements have to be 
thought out. This was done in the famous Michelson- 
Morley experiment. The result was a verdict of "death " 
to the theory of a calm ether-sea through which all 
matter moves. No dependence of the speed of light 
upon direction could be found. Not only the speed 
of light, but also other field phenomena would show 
a dependence on the direction in the moving c.s., if 
the theory of the ether-sea were assumed. Every ex- 
periment has given the same negative result as the 
Michelson-Morley one, and never revealed any de- 
pendence upon the direction of the motion of the earth. 

The situation grows more and more serious. Two 
assumptions have been tried. The first, that moving 
bodies carry ether along. The fact that the velocity of 
light does not depend on the motion of the source con- 
tradicts this assumption. The second, that there exists 
one distinguished c.s. and that moving bodies do not 
carry the ether but travel through an ever calm ether- 
sea. If this is so, then the Galilean relativity principle 
is not valid and the speed of light cannot be the same 
in every c.s. Again we are in contradiction with ex- 

More artificial theories have been tried out, assum- 
ing that the real truth lies somewhere between these 
two limiting cases : that the ether is only partially car- 
ried by the moving bodies. But they all failed ! Every 
attempt to explain the electromagnetic phenomena in 


moving c.s. with the help of the motion of the ether, 
motion through the ether, or both these motions, 
proved unsuccessful. 

Thus arose one of the most dramatic situations in 
the history of science. All assumptions concerning 
ether led nowhere ! The experimental verdict was al- 
ways negative. Looking back over the development 
of physics we see that the ether, soon after its birth, 
became the enfant terrible of the family of physical sub- 
stances. First, the construction of a simple mechanical 
picture of the ether proved to be impossible and was 
discarded. This caused, to a great extent, the break- 
down of the mechanical point of view. Second, we had 
to give up hope that through the presence of the ether- 
sea one c.s. would be distinguished and lead to the 
recognition of absolute, and not only relative, motion. 
This would have been the only way, besides carrying 
the waves, in which ether could mark and justify its 
existence. All our attempts to make ether real failed. 
It revealed neither its mechanical construction nor 
absolute motion. Nothing remained of all the pro- 
perties of the ether except that for which it was invented, 
i.e. its ability to transmit electromagnetic waves. Our 
attempts to discover the properties of the ether led to 
difficulties and contradictions. After such bad experi- 
ences, this is the moment to forget the ether completely 
and to try never to mention its name. We shall say: our 
space has the physical property of transmitting waves, 
and so omit the use of a word we have decided to avoid. 

The omission of a word from our vocabulary is, of 


course, no remedy. Our troubles are indeed much too 
profound to be solved in this way ! 

Let us now write down the facts which have been 
sufficiently confirmed by experiment without bother- 
ing any more about the "e r" problem. 

1 i ) The velocity of light in empty space always has its 
standard value, independent of the motion of the 
source or receiver of light. 

(2) In two c.s. moving uniformly, relative to each other, 
all laws of nature are exactly identical and there is 
no way of distinguishing absolute uniform motion. 

There are many experiments to confirm these two 
statements and not a single one to contradict either of 
them. The first statement expresses the constant charac- 
ter of the velocity of light, the second generalizes the 
Galilean relativity principle, formulated for mechanical 
phenomena, to all happenings in nature. 

In mechanics, we have seen : If the velocity of a ma- 
terial point is so and so, relative to one c.s., then it will 
be different in another c.s. moving uniformly, relative 
to the first. This follows from the simple mechanical 
transformation principles. They are immediately given 
by our intuition (man moving relative to ship and 
shore) and apparently nothing can be wrong here ! But 
this transformation law is in contradiction to the con- 
stant character of the velocity of light. Or, in other 
words, we add a third principle: 

(3) Positions and velocities are transformed from one 
inertial system to another according to the classical 


The contradiction is then evident. We cannot com- 
bine (i), (2), and (3). 

The classical transformation seems too obvious and 
simple for any attempt to change it. We have already 
tried to change (i) and (2) and came to a disagree- 
ment with experiment. All theories concerning the 

motion of "e r" required an alteration of (i) and 

(2). This was no good. Once more we realize the 
serious character of our difficulties. A new clue is 
needed. It is supplied by accepting the fundamental as- 
sumptions (i) and (2), and, strange though it seems, 
giving up (3). The new clue starts from an analysis of 
the most fundamental and primitive concepts ; we shall 
show how this analysis forces us to change our old 
views and removes all our difficulties. 

Our new assumptions are : 

(1) The velocity of light in vacuo is the same in all c.s. 
moving uniformly, relative to each other. 

(2) All laws of nature are the same in all c.s. moving uni- 
formly ', relative to each other. 

The relativity theory begins with these two assump- 
tions. From now on we shall not use the classical 
transformation because we know that it contradicts our 

It is essential here, as always in science, to rid our- 
selves of deep-rooted, often uncritically repeated, pre- 
judices. Since we have seen that changes in (i) and (2) 
lead to contradiction with experiment, we must have 


the courage to state their validity clearly and to attack 
the one possibly weak point, the way in which positions 
and velocities are transformed from one c.s. to another. 
It is our intention to draw conclusions from (i) and 
(2), see where and how these assumptions contradict 
the classical transformation, and find the physical 
meaning of the results obtained. 

Once more, the example of the moving room with 
outside and inside observers will be used. Again a light 
signal is emitted from the centre of the room and again 
we ask the two men what they expect to observe, as- 
suming only our two principles and forgetting what 
was previously said concerning the medium through 
which the light travels. We quote their answers: 

The inside observer: The light signal travelling from 
the centre of the room will reach the walls simultane- 
ously, since all the walls are equally distant from the 
light source and the velocity of light is the same in all 

The outside observer: In my system, the velocity of 
light is exactly the same as in that of the observer mov- 
ing with the room. It does not matter to me whether 
or not the light source moves in my c.s. since its motion 
does not influence the velocity of light. What I see is a 
light signal travelling with a standard speed, the same 
in all directions. One of the walls is trying to escape 
from and the opposite wall to approach the light signal. 
Therefore, the escaping wall will be met by the signal 
a little later than the approaching one. Although the 
difference will be very slight if the velocity of the room 


is small compared with that of light, the light signal 
will nevertheless not meet these two opposite walls, 
which are perpendicular to the direction of the motion, 
quite simultaneously. 

Comparing the predictions of our two observers, 
we find a most astonishing result which flatly contra- 
dicts the apparently well-founded concepts of classical 
physics. Two events, i.e., the two light beams reaching 
the two walls, are simultaneous for the observer on the 
inside, but not for the observer on the outside. In 
classical physics, we had one clock, one time flow, for 
all observers in all c.s. Time, and therefore such words 
as "simultaneously", " sooner", "later", had an abso- 
lute meaning independent of any c.s. Two events hap- 
pening at the same time in one c.s. happened necessarily 
simultaneously in all other c.s. 

Assumptions (i) and (2), i.e. the relativity theory, 
force us to give up this view. We have described two 
events happening at the same time in one c.s., but at 
different times in another c.s. Our task is to understand 
this consequence, to understand the meaning of the 
sentence: "Two events which are simultaneous in one 
c.s., may not be simultaneous in another c.s." 

What do we mean by "two simultaneous events in 
one c.s."? Intuitively everyone seems to know the 
meaning of this sentence. But let us make up our minds 
to be cautious and try to give rigorous definitions, as 
we know how dangerous it is to over-estimate intuition. 
Let us first answer a simple question. 

What is a clock? 


The primitive subjective feeling of time flow enables 
us to order our impressions, to judge that one event 
takes place earlier, another later. But to show that the 
time interval between two events is 10 seconds, a clock 
is needed. By the use of a clock the time concept be- 
comes objective. Any physical phenomenon may be 
used as a clock, provided it can be exactly repeated as 
many times as desired. Taking the interval between the 
beginning and the end of such an event as one unit of 
time, arbitrary time intervals may be measured by 
repetition of this physical process. All clocks, from the 
simple hour-glass to the most refined instruments, are 
based on this idea. With the hour-glass the unit of time 
is the interval the sand takes to flow from the upper to 
the lower glass. The same physical process can be re- 
peated by inverting the glass. 

At two distant points we have two perfect clocks, 
showing exactly the same time. This statement should 
be true regardless of the care with which we verify it. 
But what does it really mean? How can we make sure 
that distant clocks always show exactly the same time? 
One possible method would be to use television. It 
should be understood that television is used only as an 
example and is not essential to our argument. I could 
stand near one of the clocks and look at a televised 
picture of the other. I could then judge whether or not 
they showed the same time simultaneously. But this 
would not be a good proof. The televised picture is 
transmitted through electromagnetic waves and thus 
travels with the speed of light. Through television I 


see a picture which was sent some very short time be- 
fore, whereas on the real clock I see what is taking 
place at the present moment. This difficulty can easily 
be avoided. I must take television pictures of the two 
clocks at a point equally distant from each of them and 
observe them from this centre point. Then, if the signals 
are sent out simultaneously, they will all reach me at 
the same instant. If two good clocks observed from the 
mid-point of the distance between them always show the 
same time, then they are well suited for designating the 
time of events at two distant points. 

In mechanics we used only one clock. But this was 
not very convenient, because we had to take all measure- 
ments in the vicinity of this one clock. Looking at the 
clock from a distance, for example by television, we 
have always to remember that what we see now really 
happened earlier, just as in viewing the setting sun we 
note the event eight minutes after it has taken place. 
We should have to make corrections, according to our 
distance from the clock, in all our time readings. 

It is, therefore, inconvenient to have only one clock. 
Now, however, as we know how to judge whether 
two, or more, clocks show the same time simulta- 
neously and run in the same way, we can very well 
imagine as many clocks as we like in a given c.s. Each 
of them will help us to determine the time of the events 
happening in its immediate vicinity. The clocks are all 
at rest relative to the c.s. They are "good" clocks and 
are synchronized, which means that they show the same 
time simultaneously. 


There is nothing especially striking or strange about 
the arrangement of our clocks. We are now using 
many synchronized clocks instead of only one and can, 
therefore, easily judge whether or not two distant 
events are simultaneous in a given c.s. They are if the 
synchronized clocks in their vicinity show the same 
time at the instant the events happen. To say that one 
of the distant events happens before the other has now 
a definite meaning. All this can be judged by the help 
of the synchronized clocks at rest in our c.s. 

This is in agreement with classical physics, and not 
one contradiction to the classical transformation has 
yet appeared. 

For the definition of simultaneous events, the clocks 
are synchronized by the help of signals. It is essential 
in our arrangement that these signals travel with the 
velocity of light, the velocity which plays such a funda- 
mental role in the theory of relativity. 

Since we wish to deal with the important problem 
of two c.s. moving uniformly, relative to each other, 
we must consider two rods, each provided with clocks. 
The observer in each of the two c.s. moving relative to 
each other now has his own rod with his own set of 
clocks rigidly attached. 

When discussing measurements in classical mechanics, 
we used one clock for all c.s. Here we have many clocks 
in each c.s. This difference is unimportant. One clock 
was sufficient, but nobody could object to the use of 
many, so long as they behave as decent synchronized 
clocks should. 


Now we are approaching the essential point showing 
where the classical transformation contradicts the theory 
of relativity. What happens when two sets of clocks are 
moving uniformly, relative to each other? The classical 
physicist would answer: Nothing; they still have the 
same rhythm, and we can use moving as well as resting 
clocks to indicate time. According to classical physics, 
two events simultaneous in one c.s. will also be simul- 
taneous in any other c.s. 

But this is not the only possible answer. We can 
equally well imagine a moving clock having a different 
rhythm from one at rest. Let us now discuss this possi- 
bility without deciding, for the moment, whether or 
not clocks really change their rhythm in motion. What 
is meant by the statement that a moving clock changes 
its rhythm? Let us assume, for the sake of simplicity, 
that we have only one clock in the upper c.s. and many 
in the lower. All the clocks have the same mechanism, 
and the lower ones are synchronized, that is, they show 
the same time simultaneously. We have drawn three 
subsequent positions of the two c.s. moving relative to 
each other. In the first drawing the positions of the 
hands of the upper and lower clocks are, by conven- 
tion, the same because we arranged them so. All the 
clocks show the same time. In the second drawing, we 
see the relative positions of the two c.s. some time later. 
All the clocks in the lower c.s. show the same time, but 
the clock in the upper c.s. is out of rhythm. The rhythm 
is changed and the time differs because the clock is 
moving relative to the lower c.s. In the third drawing 


we see the difference in the positions of the hands in 
creased with time. 


An observer at rest in the lower c.s. would find that 
a moving clock changes its rhythm. Certainly the same 
result could be found if the clock moved relative to an 
observer at rest in the upper c.s.; in this case there 
would have to be many clocks in the upper c.s. and 


only one in the lower. The laws of nature must be the 
same in both c.s. moving relative to each other. 

In classical mechanics it was tacitly assumed that a 
moving clock does not change its rhythm. This seemed 
so obvious that it was hardly worth mentioning. But 
nothing should be too obvious ; if we wish to be really 
careful, we should analyse the assumptions, so far taken 
for granted, in physics. 

An assumption should not be regarded as unreason- 
able simply because it differs from that of classical 
physics. We can well imagine that a moving clock 
changes its rhythm, so long as the law of this change is 
the same for all inertial c.s. 

Yet another example. Take a yardstick; this means 
that a stick is a yard in length as long as it is at rest in 
a c.s. Now it moves uniformly, sliding along the rod 
representing the c.s. Will its length still appear to be 
one yard? We must know beforehand how to deter- 
mine its length. As long as the stick was at rest, its ends 
coincided with markings one yard apart on the c.s. 
From this we concluded : the length of the resting stick 
is one yard. How are we to measure this stick during 
motion? It could be done as follows. At a given mo- 
ment two observers simultaneously take snapshots, one 
of the origin of the stick and the other of the end. Since 
the pictures are taken simultaneously, we can compare 
the marks on the c.s. rod with which the origin and the 
end of the moving stick coincide. In this way we deter- 
mine its length. There must be two observers to take 
note of simultaneous events in different parts of the 


given c.s. There is no reason to believe that the result 
of such measurements will be the same as in the case of 
a stick at rest. Since the photographs had to be taken 
simultaneously, which is, as we already know, a relative 
concept depending on the c.s., it seems quite possible 
that the results of this measurement will be different in 
different c.s. moving relative to each other. 

We can well imagine that not only does the moving 
clock change its rhythm, but also that a moving stick 
changes its length, so long as the laws of the changes 
are the same for all inertial c.s. 

We have only been discussing some new possibilities 
without giving any justification for assuming them. 

We remember : the velocity of light is the same in all 
inertial c.s. It is impossible to reconcile this fact with 
the classical transformation. The circle must be broken 
somewhere. Can it not be done just here? Can we not 
assume such changes in the rhythm of the moving 
clock and in the length of the moving rod that the 
constancy of the velocity of light will follow directly 
from these assumptions? Indeed we can! Here is the 
first instance in which the relativity theory and classical 
physics differ radically. Our argument can be reversed : 
if the velocity of light is the same in all c.s., then mov- 
ing rods must change their length, moving clocks must 
change their rhythm, and the laws governing these 
changes are rigorously determined. 

There is nothing mysterious or unreasonable in all 
this. In classical physics it was always assumed that 
clocks in motion and at rest have the same rhythm, that 



rods in motion and at rest have the same length. If the 
velocity of light is the same in all c.s., if the relativity 
theory is valid, then we must sacrifice this assumption. 
It is difficult to get rid of deep-rooted prejudices, but 
there is no other way. From the point of view of the 
relativity theory the old concepts seem arbitrary. Why 
believe, as we did some pages ago, in absolute time 
flowing in the same way for all observers in all c.s.? 
Why believe in unchangeable distance? Time is deter- 
mined by clocks, space co-ordinates by rods, and the 
result of their determination may depend on the be- 
haviour of these clocks and rods when in motion. There 
is no reason to believe that they will behave in the way 
we should like them to. Observation shows, indirectly, 
through the phenomena of electromagnetic field, that 
a moving clock changes its rhythm, a rod its length, 
whereas on the basis of mechanical phenomena we did 
not think this happened. We must accept the concept 
of relative time in every c.s., because it is the best way 
out of our difficulties. Further scientific advance, de- 
veloping from the theory of relativity, shows that this 
new aspect should not be regarded as a malum necessarium, 
for the merits of the theory are much too marked. 

So far we have tried to show what led to the funda- 
mental assumptions of the relativity theory, and how 
the theory forced us to revise and to change the classical 
transformation by treating time and space in a new 
way. Our aim is to indicate the ideas forming the basis 
of a new physical and philosophical view. These ideas 
are simple; but in the form in which they have been 


formulated here, they are insufficient for arriving at 
not only qualitative, but also quantitative conclusions. 
We must again use our old method of explaining only 
the principal ideas and stating some of the others with- 
out proof. 

To make clear the difference between the view of 
the old physicist, whom we shall call and who 
believes in the classical transformation, and that of the 
modern physicist, whom we shall call M and who 
knows the relativity theory, we shall imagine a dialogue 
between them. 

0. I believe in the Galilean relativity principle in 
mechanics, because I know that the laws of mechanics 
are the same in two c.s. moving uniformly relative to 
each other, or in other words, that these laws are in- 
variant with respect to the classical transformation. 

M . But the relativity principle must apply to all events 
in our external world. Not only the laws of mechanics 
but all laws of nature must be the same in c.s. moving 
uniformly, relative to each other. 

0. But how can all laws of nature possibly be the 
same in c.s. moving relative to each other? The field 
equations, that is, Maxwell's equations, are not in- 
variant with respect to the classical transformation. This 
is clearly shown by the example of the velocity of light. 
According to the classical transformation, this velocity 
should not be the same in two c.s. moving relative to 
each other. 

M. This merely shows that the classical transforma- 
tion cannot be applied, that the connection between 


two c.s. must be different; that we may not connect 
co-ordinates and velocities as is done in these trans- 
formation laws. We have to substitute new laws and 
deduce them from the fundamental assumptions of the 
theory of relativity. Let us not bother about the mathe- 
matical expression for this new transformation law, 
and be satisfied that it is different from the classical. 
We shall call it briefly the Lorentz transformation. It 
can be shown that Maxwell's equations, that is, the 
laws of field, are invariant with respect to the Lorentz 
transformation, just as the laws of mechanics are in- 
variant with respect to the classical transformation. 
Remember how it was in classical physics. We had 
transformation laws for co-ordinates, transformation 
laws for velocities, but the laws of mechanics were the 
same for two c.s. moving uniformly, relative to each 
other. We had transformation laws for space, but not 
for time, because time was the same in all c.s. Here, 
however, in the relativity theory, it is different. We 
have transformation laws different from the classical 
for space, time, and velocity. But again the laws of 
nature must be the same in all c.s. moving uniformly, 
relative to each other. The laws of nature must be 
invariant, not, as before, with respect to the classical 
transformation, but with respect to a new type of 
transformation, the so-called Lorentz transformation. 
In all inertial c.s. the same laws are valid and the tran- 
sition from one c.s. to another is given by the Lorentz 

0. I take your word for it, but it would interest me 


to know the difference between the classical and Lorentz 
transformations . 

M. Your question is best answered in the following 
way. Quote some of the characteristic features of the 
classical transformation and I shall try to explain 
whether or not they are preserved in the Lorentz 
transformation, and if not, how they are changed. 

0. If something happens at some point at some time 
in my c.s., then the observer in another c.s. moving 
uniformly, relative to mine, assigns a different number 
to the position in which this event occurs, but of course 
the same time. We use the same clock in all our c.s. 
and it is immaterial whether or not the clock moves. Is 
this also true for you? 

M. No, it is not. Every c.s. must be equipped with 
its own clocks at rest, since motion changes the rhythm. 
Two observers in two different c.s. will assign not only 
different numbers to the position, but also different 
numbers to the time at which this event happens. 

0. This means that the time is no longer an in- 
variant. In the classical transformation it is always the 
same time in all c.s. In the Lorentz transformation it 
changes and somehow behaves like the co-ordinate in the 
old transformation. I wonder how it is with distance? 
According to classical mechanics a rigid rod preserves 
its length in motion or at rest. Is this also true now? 

M. It is not. In fact, it follows from the Lorentz 
transformation that a moving stick contracts in the 
direction of the motion and the contraction increases if 
the speed increases. The faster a stick moves, the shorter 


it appears. But this occurs only in the direction of the 
motion. You see in my drawing a moving rod which 
shrinks to half its length, when it moves with a velocity 
approaching ca. 90 per cent of the velocity of light. 
There is no contraction, however, in the direction 
perpendicular to the motion, as I have tried to illustrate 
in my drawing. 

0. This means that the rhythm of a moving clock 
and the length of a moving stick depend on the speed. 
But how? 

M. The changes become more distinct as the speed 
increases. It follows from the Lorentz transformation 
that a stick would shrink to nothing if its speed were to 




reach that of light. Similarly the rhythm of a moving 
clock is slowed down, compared to the clocks it passes 
along the rod, and would come to a stop if the clock 
were to move with the speed of light, that is, if the clock 
is a "good" one. 

0. This seems to contradict all our experience. We 
know that a car does not become shorter when in 
motion and we also know that the driver can always 
compare his "good" watch with those he passes on the 
way, finding that they agree fairly well, contrary to 
your statement. 

M. This is certainly true. But these mechanical 
velocities are all very small compared to that of light, 
and it is, therefore, ridiculous to apply relativity to 
these phenomena. Every car driver can safely apply 
classical physics even if he increases his speed a hundred 
thousand times. We could only expect disagreement 
between experiment and the classical transformation 
with velocities approaching that of light. Only with very 
great velocities can the validity of the Lorentz trans- 
formation be tested. 

0. But there is yet another difficulty. According to 
mechanics I can imagine bodies with velocities even 
greater than that of light. A body moving with the 
velocity of light relative to a floating ship moves with a 
velocity greater than that of light relative to the shore. 
What will happen to the stick which shrank to nothing 
when its velocity was that of light? We can hardly 
expect a negative length if the velocity is greater than 
that of light. 


M. There is really no reason for such sarcasm ! From 
the point of view of the relativity theory a material 
body cannot have a velocity greater than that of light. 
The velocity of light forms the upper limit of veloci- 
ties for all material bodies. If the speed of a body is 
equal to that of light relative to a ship, then it will 
also be equal to that of light relative to the shore. 
The simple mechanical law of adding and subtracting 
velocities is no longer valid or, more precisely, is only 
approximately valid for small velocities, but not for 
those near the velocity of light. The number expressing 
the velocity of light appears explicitly in the Lorentz 
transformation, and plays the role of a limiting case, 
like the infinite velocity in classical mechanics. This 
more general theory does not contradict the classical 
transformation and classical mechanics. On the con- 
trary, we regain the old concepts as a limiting case when 
the velocities are small. From the point of view of the 
new theory it is clear in which cases classical physics 
is valid and wherein its limitations lie. It would be just 
as ridiculous to apply the theory of relativity to the 
motion of cars, ships, and trains as to use a calculating 
machine where a multiplication table would be suffi- 


The relativity theory arose from necessity, from serious 
and deep contradictions in the old theory from which 
there seemed no escape. The strength of the new theory 
lies in the consistency and simplicity with which it 


solves all these difficulties, using only a few very con- 
vincing assumptions. 

Although the theory arose from the field problem, it 
has to embrace all physical laws. A difficulty seems to 
appear here. The field laws on the one hand and the 
mechanical laws on the other are of quite different 
kinds. The equations of electromagnetic field are in- 
variant with respect to the Lorentz transformation and 
the mechanical equations are invariant with respect to 
the classical transformation. But the relativity theory 
claims that all laws of nature must be invariant with 
respect to the Lorentz and not to the classical trans- 
formation. The latter is only a special, limiting case of 
the Lorentz transformation when the relative velocities 
of two c.s. are very small. If this is so, classical mechanics 
must change in order to conform with the demand of 
invariance with respect to the Lorentz transformation. 
Or, in other words, classical mechanics cannot be valid 
if the velocities approach that of light. Only one trans- 
formation from one c.s. to another can exist, namely, 
the Lorentz transformation. 

It was simple to change classical mechanics in such a 
way that it contradicted neither the relativity theory 
nor the wealth of material obtained by observation and 
explained by classical mechanics. The old mechanics is 
valid for small velocities and forms the limiting case of 
the new one. 

It would be interesting to consider some instance 
of a change in classical mechanics introduced by the 
relativity theory. This might, perhaps, lead us to some 


conclusions which could be proved or disproved by 

Let us assume a body, having a definite mass, moving 
along a straight line, and acted upon by an external 
force in the direction of the motion. The force, as we 
know, is proportional to the change of velocity. Or, to 
be more explicit, it does not matter whether a given 
body increases its velocity in one second from 100 to 
101 feet per second, or from 100 miles to 100 miles and 
one foot per second or from 180,000 miles to 180,000 
miles and one foot per second. The force acting upon 
a particular body is always the same for the same change 
of velocity in the same time. 

Is this sentence true from the point of view of the 
relativity theory? By no means ! This law is valid only 
for small velocities. What, according to the relativity 
theory, is the law for great velocities, approaching that 
of light? If the velocity is great, extremely strong forces 
are required to increase it. It is not at all the same 
thing to increase by one foot per second a velocity 
of about 100 feet per second or a velocity approach- 
ing that of light. The nearer a velocity is to that of 
light the more difficult it is to increase. When a 
velocity is equal to that of light it is impossible to 
increase it further. Thus, the changes brought about 
by the relativity theory are not surprising. The velocity 
of light is the upper limit for all velocities. No finite 
force, no matter how great, can cause an increase in 
speed beyond this limit. In place of the old mechanical 
law connecting force and change of velocity, a more 


complicated one appears. From our new point of view 
classical mechanics is simple because in nearly all 
observations we deal with velocities much smaller than 
that of light. 

A body at rest has a definite mass, called the rest mass. 
We know from mechanics that every body resists a 
change in its motion; the greater the mass, the stronger 
the resistance, and the weaker the mass, the weaker the 
resistance. But in the relativity theory, we have some- 
thing more. Not only does a body resist a change more 
strongly if the rest mass is greater, but also if its velocity 
is greater. Bodies with velocities approaching that of 
light would offer a very strong resistance to external 
forces. In classical mechanics the resistance of a given 
body was something unchangeable, characterized by its 
mass alone. In the relativity theory it depends on both 
rest mass and velocity. The resistance becomes infinitely 
great as the velocity approaches that of light. 

The results just quoted enable us to put the theory to 
the test of experiment. Do projectiles with a velocity 
approaching that of light resist the action of an external 
force as predicted by the theory? Since the statements 
of the relativity theory have, in this respect, a quantita- 
tive character, we could confirm or disprove the theory 
if we could realize projectiles having a speed approaching 
that of light. 

Indeed, we find in nature projectiles with such veloci- 
ties. Atoms of radioactive matter, radium for instance, 
act as batteries which fire projectiles with enormous 
velocities. Without going into detail we can quote only 


one of the very important views of modern physics 
and chemistry. All matter in the universe is made 
up of elementary particles of only a few kinds. It is like 
seeing in one town buildings of different sizes, construc- 
tion and architecture, but from shack to skyscraper 
only very few different kinds of bricks were used, the 
same in all the buildings. So all known elements of 
our material world from hydrogen the lightest, to 
uranium the heaviest are built of the same kinds of 
bricks, that is, the same kinds of elementary particles. 
The heaviest elements, the most complicated buildings, 
are unstable and they disintegrate or, as we say, are 
radioactive. Some of the bricks, that is, the elementary 
particles of which the radioactive atoms are con- 
structed, are sometimes thrown out with a very great 
velocity, approaching that of light. An atom of an 
element, say radium, according to our present views, 
confirmed by numerous experiments, is a complicated 
structure, and radioactive disintegration is one of 
those phenomena in which the composition of atoms 
from still simpler bricks, the elementary particles, is 

By very ingenious and intricate experiments we can 
find out how the particles resist the action of an external 
force. The experiments show that the resistance offered 
by these particles depends on the velocity, in the way 
foreseen by the theory of relativity. In many other 
cases, where the dependence of the resistance upon the 
velocity could be detected, there was complete agree- 
ment between theory and experiment. We see once 


more the essential features of creative work in science : 
prediction of certain facts by theory and their con- 
firmation by experiment. 

This result suggests a further important generaliza- 
tion. A body at rest has mass but no kinetic energy, 
that is, energy of motion. A moving body has both 
mass and kinetic energy. It resists change of velocity 
more strongly than the resting body. It seems as though 
the kinetic energy of the moving body increases its 
resistance. If two bodies have the same rest mass, the 
one with the greater kinetic energy resists the action of 
an external force more strongly. 

Imagine a box containing balls, with the box as well 
as the balls at rest in our c.s. To move it, to increase its 
velocity, some force is required. But will the same 
force increase the velocity by the same amount in the 
same time with the balls moving about quickly and in 
all directions inside the box, like the molecules of a gas, 
with an average speed approaching that of light? A 
greater force will now be necessary because of the 
increased kinetic energy of the balls, strengthening the 
resistance of the box. Energy, at any rate kinetic energy, 
resists motion in the same way as ponderable masses. Is 
this also true of all kinds of energy? 

The theory of relativity deduces, from its funda- 
mental assumption, a clear and convincing answer to 
this question, an answer again of a quantitative charac- 
ter: all energy resists change of motion; all energy 
behaves like matter; a piece of iron weighs more when 
red-hot than when cool; radiation travelling through 


space and emitted from the sun contains energy and 
therefore has mass; the sun and all radiating stars lose 
mass by emitting radiation. This conclusion, quite 
general in character, is an important achievement of 
the theory of relativity and fits all facts upon which it 
has been tested. 

Classical physics introduced two substances: matter 
and energy. The first had weight, but the second was 
weightless. In classical physics we had two conserva- 
tion laws: one for matter, the other for energy. We 
have already asked whether modern physics still holds 
this view of two substances and the two conservation 
laws. The answer is: "No". According to the theory 
of relativity, there is no essential distinction between 
mass and energy. Energy has mass and mass represents 
energy. Instead of two conservation laws we have only 
one, that of mass-energy. This new view proved very 
successful and fruitful in the further development of 

How is it that this fact of energy having mass and 
mass representing energy remained for so long ob- 
scured? Is the weight of a piece of hot iron greater than 
that of a cold piece? The answer to this question is now 
"Yes", but on p. 43 it was "No". The pages between 
these two answers are certainly not sufficient to cover 
this contradiction. 

The difficulty confronting us here is of the same kind 
as we have met before. The variation of mass predicted 
by the theory of relativity is immeasurably small and 
cannot be detected by direct weighing on even the most 


sensitive scales. The proof that energy is not weightless 
can be gained in many very conclusive, but indirect, 

The reason for this lack of immediate evidence is the 
very small rate of exchange between matter and energy. 
Compared to mass, energy is like a depreciated currency 
compared to one of high value. An example will make 
this clear. The quantity of heat able to convert thirty 
thousand tons of water into steam would weigh about 
one gram ! Energy was regarded as weightless for so 
long simply because the mass which it represents is so 

The old energy-substance is the second victim of the 
theory of relativity. The first was the medium through 
which light waves were propagated. 

The influence of the theory of relativity goes far 
beyond the problem from which it arose. It removes 
the difficulties and contradictions of the field theory; 
it formulates more general mechanical laws ; it replaces 
two conservation laws by one; it changes our classical 
concept of absolute time. Its validity is not restricted 
to one domain of physics ; it forms a general framework 
embracing all phenomena of nature. 


"The French revolution began in Paris on the 1/ 
of July 1789." In this sentence the place and time of 
an event are stated. Hearing this statement for the first 
time, one who does not know what " Paris" means could 
be taught: it is a city on our earth situated in long. 2 


East and lat. 49 North. The two numbers would then 
characterize the place, and "i4th of July 1789" the 
time, at which the event took place. In physics, much 
more than in history, the exact characterization of when 
and where an event takes place is very important, 
because these data form the basis for a quantitative 

For the sake of simplicity, we considered previously 
only motion along a straight line. A rigid rod with an 
origin but no end-point was our c.s. Let us keep this 
restriction. Take different points on the rod; their 
positions can be characterized by one number only, by 
the co-ordinate of the point. To say the co-ordinate of 
a point is 7*586 feet means that its distance is 7*586 feet 
from the origin of the rod. If, on the contrary, some- 
one gives me any number and a unit, I can always find 
a point on the rod corresponding to this number. We 
can state: a definite point on the rod corresponds to 
every number, and a definite number corresponds to 
every point. This fact is expressed by mathematicians 
in the following sentence: all points on the rod form 
a one-dimensional continuum. There exists a point arbi- 
trarily near every point on the rod. We can connect 
two distant points on the rod by steps as small as we 
wish. Thus the arbitrary smallness of the steps con- 
necting distant points is characteristic of the con- 

Now another example. We have a plane, or, if you 
prefer something more concrete, the surface of a rect- 
angular table. The position of a point on this table 


can be characterized by two numbers and not, as before, 
by one. The two numbers are the distances from two 
perpendicular edges of the table. Not one number, 
but a pair of numbers corresponds to every point on 
the plane; a definite point corresponds to every pair of 
numbers. In other words : the plane is a two-dimensional 
continuum. There exist points arbitrarily near every point 
on the plane. Two distant points can be connected by 
a curve divided into steps as small as we wish. Thus 

the arbitrary smallness of the steps connecting two 
distant points, each of which can be represented by two 
numbers, is again characteristic of a two-dimensional 

One more example. Imagine that you wish to regard 
your room as your c.s. This means that you want to 
describe all positions with respect to the rigid walls 
of the room. The position of the end-point of the 
lamp, if the lamp is at rest, can be described by three 
numbers: two of them determine the distance from 
two perpendicular walls, and the third that from the 
floor or ceiling. Three definite numbers correspond to 
every point of the space; a definite point in space 


corresponds to every three numbers. This is expressed 
by the sentence: Our space is a three-dimensional con- 
tinuum. There exist points very near every point of 
the space. Again the arbitrary smallness of the steps 
connecting the distant points, each of them represented 
by three numbers, is characteristic of a three-dimensional 

But all this is scarcely physics. To return to physics, 
the motion of material particles must be considered. 
To observe and predict events in nature we must con- 
sider not only the place but also the time of physical 
happenings. Let us again take a very simple ex- 

A small stone, which can be regarded as a particle, 
is dropped from a tower. Imagine the tower 256 feet 
high. Since Galileo's time we have been able to 
predict the co-ordinate of the stone at any arbitrary 
instant after it was dropped. Here is a "timetable" 
describing the positions of the stone after o, 1,2, 3, and 
4 seconds. 



Time in 

Elevation from 
the ground in feet 







Five events are registered in our "timetable", each 
represented by two numbers, the time and space co- 
ordinates of each event. The first event is the dropping 
of the stone from 256 feet above the ground at the zero 
second. The second event is the coincidence of the 
stone with our rigid rod (the tower) at 240 feet above 
the ground. This happens after the first second. The 
last event is the coincidence of the stone with the 

We could represent the knowledge gained from our 
"timetable" in a different way. We could represent the 
five pairs of numbers in the " timetable" as five points 
on a surface. Let us first establish a scale. One segment 
will correspond to a foot and another to a second. For 
example : 

100 Ft. 


We then draw two perpendicular lines, calling the 
horizontal one, say, the time axis and the vertical one 
the space axis. We see immediately that our "time- 
table" can be represented by five points in our time- 
space plane. 


The distances of the points from the space axis repre- 
sent the time co-ordinates as registered in the first 
column of our "timetable", and the distances from the 
time axis their space co-ordinates. 




2 5 4 

Time axis 


Exactly the same thing is expressed in two different 
ways: by the " timetable 5 ' and by the points on the 
plane. Each can be constructed from the other. The 
choice between these two representations is merely a 
matter of taste, for they are, in fact, equivalent. 

Let us now go one step farther. Imagine a better 
c 'timetable" giving the positions not for every second, 
but for, say, every hundredth or thousandth of a second. 
We shall then have very many points on our time-space 
plane. Finally, if the position is given for every instant 
or, as the mathematicians say, if the space co-ordinate 


is given as a function of time, then our set of points 
becomes a continuous line. Our next drawing therefore 
represents not just a fragment as before, but a complete 
knowledge of the motion. 



Time axis 

The motion along the rigid rod (the tower), the 
motion in a one-dimensional space, is here represented 
as a curve in a two-dimensional time-space continuum. 
To every point in our time-space continuum there 
corresponds a pair of numbers, one of which denotes 
the time, and the other the space, co-ordinate. Con- 
versely: a definite point in our time-space plane corre- 
sponds to every pair of numbers characterizing an event. 
Two adjacent points represent two events, two happen- 
ings, at slightly different places and at slightly different 

You could argue against our representation thus: 


there is little sense in representing a unit of time by a 
segment, in combining it mechanically with the space, 
forming the two-dimensional continuum from the two 
one-dimensional continua. But you would then have 
to protest just as strongly against all the graphs repre- 
senting, for example, the change of temperature in 
New York City during last summer, or against those 
representing the changes in the cost of living during 
the last few years, since the very same method is used 
in each of these cases. In the temperature graphs the 
one-dimensional temperature continuum is combined 
with the one-dimensional time continuum into the 
two-dimensional temperature-time continuum. 

Let us return to the particle dropped from a 256-foot 
tower. Our graphic picture of motion is a useful con- 
vention since it characterizes the position of the particle 
at an arbitrary instant. Knowing how the particle 
moves, we should like to picture its motion once more. 
We can do this in two different ways. 

We remember the picture of the particle changing 
its position with time in the one-dimensional space. 
We picture the motion as a sequence of events in the 
one-dimensional space continuum. We do not mix time 
and space, using a dynamic picture in which positions 
change with time. 

But we can picture the same motion in a different 
way. We can form a static picture, considering the curve 
in the two-dimensional time-space continuum. Now 
the motion is represented as something which is, which 
exists in the two-dimensional tim^space continuum, 


and not as something which changes in the one-dimen- 
sional space continuum. 

Both these pictures are exactly equivalent, and pre- 
ferring one to the other is merely a matter of convention 
and taste. 

Nothing that has been said here about the two 
pictures of the motion has anything whatever to do 
with the relativity theory. Both representations can be 
used with equal right, though classical physics favoured 
rather the dynamic picture describing motion as happen- 
ings in space and not as existing in time-space. But the 
relativity theory changes this view. It was distinctly in 
favour of the static picture and found in this representa- 
tion of motion as something existing in time-space a 
more convenient and more objective picture of reality. 
We still have to answer the question : why are these two 
pictures, equivalent from the point of view of classical 
physics, not equivalent from the point of view of the 
relativity theory? 

The answer will be understood if two c.s. moving 
uniformly, relative to each other, are again taken into 

According to classical physics, observers in two c.s. 
moving uniformly, relative to each other, will assign 
different space co-ordinates, but the same time co- 
ordinate, to a certain event. Thus in our example, the 
coincidence of the particle with the earth is character- 
ized in our chosen c.s. by the time co-ordinate "4" and 
by the space co-ordinate "o". According to classical 
mechanics, the stone will still reach the earth after 


four seconds for an observer moving uniformly, relative 
to the chosen c.s. But this observer will refer the distance 
to his c.s. and will, in general, connect different space 
co-ordinates with the event of collision, although the 
time co-ordinate will be the same for him and for all 
other observers moving uniformly, relative to each other. 
Classical physics knows only an "absolute" time flow 
for all observers. For every c.s. the two-dimensional 
continuum can be split into two one-dimensional con- 
tinua: time and space. Because of the "absolute" 
character of time, the transition from the "static" to 
the "dynamic" picture of motion has an objective 
meaning in classical physics. 

But we have already allowed ourselves to be convinced 
that the classical transformation must not be used in 
physics generally. From a practical point of view it is 
still good for small velocities, but not for settling funda- 
mental physical questions. 

According to the relativity theory the time of the 
collision of the stone with the earth will not be the 
same for all observers. The time co-ordinate and the 
space co-ordinate will be different in two c.s., and the 
change in the time co-ordinate will be quite distinct 
if the relative velocity is close to that of light. The 
two-dimensional continuum cannot be split into two 
one-dimensional continua as in classical physics. We 
must not consider space and time separately in deter- 
mining the time-space co-ordinates in another c.s. The 
splitting of the two-dimensional continuum into two 
one-dimensional ones seems, from the point of view 


of the relativity theory, to be an arbitrary procedure 
without objective meaning. 

It will be simple to generalize all that we have just 
said for the case of motion not restricted to a straight 
line. Indeed, not two, but four, numbers must be used 
to describe events in nature. Our physical space as 
conceived through objects and their motion has three 
dimensions, and positions are characterized by three 
numbers. The instant of an event is the fourth number. 
Four definite numbers correspond to every event; a 
definite event corresponds to any four numbers. There- 
fore: The world of events forms a four-dimensional 
continuum. There is nothing mysterious about this, 
and the last sentence is equally true for classical 
physics and the relativity theory. Again, a difference 
is revealed when two c.s. moving relatively to each 
other are considered. The room is moving, and the 
observers inside and outside determine the time-space 
co-ordinates of the same events. Again, the classical 
physicist splits the four-dimensional continua into the 
three-dimensional spaces and the one-dimensional 
time-continuum. The old physicist bothers only about 
space transformation, as time is absolute for him. He 
finds the splitting of the four-dimensional world-continua 
into space and time natural and convenient. But from the 
point of view of the relativity theory, time as well as 
space is changed by passing from one c.s. to another, 
and the Lorentz transformation considers the trans- 
formation properties of the four-dimensional time-space 
continuum of our four-dimensional world of events. 


The world of events can be described dynamically 
by a picture changing in time and thrown on to the 
background of the three-dimensional space. But it can 
also be described by a static picture thrown on to the 
background of a four-dimensional time-space con- 
tinuum. From the point of view of classical physics the 
two pictures, the dynamic and the static, are equivalent. 
But from the point of view of the relativity theory the 
static picture is the more convenient and the more 

Even in the relativity theory we can still use the 
dynamic picture if we prefer it. But we must remem- 
ber that this division into time and space has no 
objective meaning since time is no longer " absolute". 
We shall still use the "dynamic" and not the "static" 
language in the following pages, bearing in mind its 


There still remains one point to be cleared up. One of 
the most fundamental questions has not been settled as 
yet: does an inertial system exist? We have learned 
something about the laws of nature, their invariance 
with respect to the Lorentz transformation, and their 
validity for all inertial systems moving uniformly, 
relative to each other. We have the laws but do not 
know the frame to which to refer them. 

In order to be more aware of this difficulty, let us 
interview the classical physicist and ask him some 
simple questions : 


"What is an inertial system? 5 ' 

" It is a c.s. in which the laws of mechanics are valid. 
A body on which no external forces are acting moves 
uniformly in such a c.s. This property thus enables us 
to distinguish an inertial c.s. from any other." 

"But what does it mean to say that no forces are 
acting on a body?" 

"It simply means that the body moves uniformly in 
an inertial c.s." 

Here we could once more put the question: "What 
is an inertial c.s.?" But since there is little hope of ob- 
taining an answer differing from the above, let us try to 
gain some concrete information by changing the question: 

"Is a c.s. rigidly connected with the earth an inertial 

"No, because the laws of mechanics are not rigor- 
ously valid on the earth, due to its rotation. A c.s. 
rigidly connected with the sun can be regarded for 
many problems as an inertial c.s. ; but when we speak 
of the rotating sun, we again understand that a c.s. 
connected with it cannot be regarded as strictly 

"Then what, concretely, is your inertial c.s., and 
how is its state of motion to be chosen?" 

"It is merely a useful fiction and I have no idea how 
to realize it. If I could only get far away from all 
material bodies and free myself from all external in- 
fluences, my c.s. would then be inertial." 

"But what do you mean by a c.s. free from all 
external influences?" 


"I mean that the c.s. is inertial." 

Once more we are back at our initial question ! 

Our interview reveals a grave difficulty in classical 
physics. We have laws, but do not know what frame to 
refer them to, and our whole physical structure seems 
to be built on sand. 

We can approach this same difficulty from a different 
point of view. Try to imagine that there is only one 
body, forming our c.s., in the entire universe. This body 
begins to rotate. According to classical mechanics, the 
physical laws for a rotating body are different from 
those for a non-rotating body. If the inertial principle 
is valid in one case, it is not valid in the other. But all 
this sounds very suspicious. Is it permissible to consider 
the motion of only one body in the entire universe? 
By the motion of a body we always mean its change 
of position in relation to a second body. It is, therefore, 
contrary to common sense to speak about the motion 
of only one body. Classical mechanics and common 
sense disagree violently on this point. Newton's recipe 
is: if the inertial principle is valid, then the c.s. is either 
at rest or in uniform motion. If the inertial principle 
is invalid, then the body is in non-uniform motion. 
Thus, our verdict of motion or rest depends upon 
whether or not all the physical laws are applicable to 
a given c.s. 

Take two bodies, the sun and the earth, for instance. 
The motion we observe is again relative. It can be 
described by connecting the c.s. with either the earth 
or the sun. From this point of view, Copernicus' great 


achievement lies in transferring the c.s. from the earth 
to the sun. But as motion is relative and any frame of 
reference can be used, there seems to be no reason for 
favouring one c.s. rather than the other. 

Physics again intervenes and changes our common- 
sense point of view. The c.s. connected with the sun 
resembles an inertial system more than that connected 
with the earth. The physical laws should be applied to 
Copernicus' c.s. rather than to Ptolemy's. The greatness 
of Copernicus' discovery can be appreciated only from 
the physical point of view. It illustrates the great 
advantage of using a c.s. connected rigidly with the sun 
for describing the motion of planets. 

No absolute uniform motion exists in classical physics. 
If two c.s. are moving uniformly, relative to each other, 
then there is no sense in saying, "This c.s. is at rest and 
the other is moving". But if two c.s. are moving non- 
uniformly, relative to each other, then there is very 
good reason for saying, "This body moves and the 
other is at rest (or moves uniformly) ". Absolute motion 
has here a very definite meaning. There is, at this 
point, a wide gulf between common sense and classical 
physics. The difficulties mentioned, that of an inertial 
system and that of absolute motion, are strictly con- 
nected with each other. Absolute motion is made 
possible only by the idea of an inertial system, for which 
the laws of nature are valid. 

It may seem as though there is no way out of these 
difficulties, as though no physical theory can avoid 
them. Their root lies in the validity of the laws of nature 


for a special class of c.s. only, the inertial. The possibility 
of solving these difficulties depends on the answer to 
the following question. Can we formulate physical 
laws so that they are valid for all c.s., not only those 
moving uniformly, but also those moving quite arbi- 
trarily, relative to each other? If this can be done, 
our difficulties will be over. We shall then be able to 
apply the laws of nature to any c.s. The struggle, so 
violent in the early days of science, between the views 
of Ptolemy and Copernicus would then be quite mean- 
ingless. Either c.s. could be used with equal justification. 
The two sentences, u the sun is at rest and the earth 
moves", or "the sun moves and the earth is at rest", 
would simply mean two different conventions concern- 
ing two different c.s. 

Could we build a real relativistic physics valid in all 
c.s.; a physics in which there would be no place for 
absolute, but only for relative, motion? This is indeed 
possible ! 

We have at least one indication, though a very weak 
one, of how to build the new physics. Really relativistic 
physics must apply to all c.s. and, therefore, also to the 
special case of the inertial c.s. We already know the 
laws for this inertial c.s. The new general laws valid for 
all c.s. must, in the special case of the inertial system, 
reduce to the old, known laws. 

The problem of formulating physical laws for every 
c.s. was solved by the so-called general relativity theory \ 
the previous theory, applying only to inertial systems, 
is called the special relativity theory. The two theories 


cannot, of course, contradict each other, since we 
must always include the old laws of the special 
relativity theory in the general laws for an inertial 
system. But just as the inertial c.s. was previously the 
only one for which physical laws were formulated, so now 
it will form the special limiting case, as all c.s. moving 
arbitrarily, relative to each other, are permissible. 

This is the programme for the general theory of 
relativity. But in sketching the way in which it was 
accomplished we must be even vaguer than we have 
been so far. New difficulties arising in the development 
of science force our theory to become more and more 
abstract. Unexpected adventures still await us. But 
our final aim is always a better understanding of reality. 
Links are added to the chain of logic connecting theory 
and observation. To clear the way leading from theory 
to experiment of unnecessary and artificial assumptions, 
to embrace an ever-wider region of facts, we must 
make the chain longer and longer. The simpler and 
more fundamental our assumptions become, the more 
intricate is our mathematical tool of reasoning; the 
way from theory to observation becomes longer, more 
subtle, and more complicated. Although it sounds 
paradoxical, we could say: Modern physics is simpler 
than the old physics and seems, therefore, more difficult 
and intricate. The simpler our picture of the external 
world and the more facts it embraces, the more 
strongly it reflects in our minds the harmony of the 

Our new idea is simple: to build a physics valid for 


all c.s. Its fulfilment brings formal complications and 
forces us to use mathematical tools different from those 
so far employed in physics. We shall show here only the 
connection between the fulfilment of this programme 
and two principal problems : gravitation and geometry. 


The law of inertia marks the first great advance in 
physics; in fact, its real beginning. It was gained by 
the contemplation of an idealized experiment, a body 
moving forever with no friction nor any other external 
forces acting. From this example, and later from many 
others, we recognized the importance of the idealized 
experiment created by thought. Here again, idealized 
experiments will be discussed. Although these may 
sound very fantastic, they will, nevertheless, help us to 
understand as much about relativity as is possible by 
our simple methods. 

We had previously the idealized experiments with a 
uniformly moving room. Here, for a change, we shall 
have a falling lift. 

Imagine a great lift at the top of a skyscraper 
much higher than any real one. Suddenly the cable 
supporting the lift breaks, and the lift falls freely 
toward the ground. Observers in the lift are per- 
forming experiments during the fall. In describing 
them, we need not bother about air resistance or 
friction, for we may disregard their existence under 
our idealized conditions. One of the observers takes a 
handkerchief and a watch from his pocket and drops 


them. What happens to these two bodies? For the 
outside observer, who is looking through the window of 
the lift, both handkerchief and watch fall toward the 
ground in exactly the same way, with the same accelera- 
tion. We remember that the acceleration of a falling 
body is quite independent of its mass and that it was 
this fact which revealed the equality of gravitational 
and inertial mass (p. 37). We also remember that the 
equality of the two masses, gravitational and inertial, 
was quite accidental from the point of view of classical 
mechanics and played no role in its structure. Here, 
however, this equality reflected in the equal acceleration 
of all falling bodies is essential and forms the basis of 
our whole argument. 

Let us return to o\ir falling handkerchief and watch; 
for the outside observer they are both falling with the 
same acceleration. But so is the lift, with its walls, 
ceiling, and floor. Therefore : the distance between the 
two bodies and the floor will not change. For the 
inside observer the two bodies remain exactly where 
they were when he let them go. The inside observer 
may ignore the gravitational field, since its source lies 
outside his c.s. He finds that no forces inside the lift 
act upon the two bodies, and so they are at rest, 
just as if they were in an inertial c.s. Strange things 
happen in the lift ! If the observer pushes a body 
in any direction, up or down for instance, it always 
moves uniformly so long as it does not collide with 
the ceiling or the floor of the lift. Briefly speaking, 
the laws of classical mechanics are valid for the observer 



inside the lift. All bodies behave in the way expected 
by the law of inertia. Our new c.s. rigidly connected 
with the freely falling lift differs from the inertial 
c.s. in only one respect. In an inertial c.s., a moving 
body on which no forces are acting will move uni- 
formly for ever. The inertial c.s. as represented in 
classical physics is neither limited in space nor time. 
The case of the observer in our lift is, however, 
different. The inertial character of his c.s. is limited 
in space and time. Sooner or later the uniformly 
moving body will collide with the wall of the lift, 
destroying the uniform motion. Sooner or later the 
whole lift will collide with the earth, destroying the 
observers and their experiments. The c.s. is only a 
"pocket edition" of a real inertial c.s. 

This local character of the c.s. is quite essential. If 
our imaginary lift were to reach from the North Pole 
to the Equator, with the handkerchief placed over the 
North Pole and the watch over the Equator, then, for 
the outside observer, the two bodies would not have the 
same acceleration; they would not be at rest relative 
to each other. Our whole argument would fail ! The 
dimensions of the lift must be limited so that the equality 
of acceleration of all bodies relative to the outside 
observer may be assumed. 

With this restriction, the c.s. takes on an inertial 
character for the inside observer. We can at least indi- 
cate a c.s. in which all the physical laws are valid, even 
though it is limited in time and space. If we imagine 
another c.s., another lift moving uniformly, relative to 


the one falling freely, then both these c.s. will be locally 
inertial. All laws are exactly the same in both. The 
transition from one to the other is given by the Lorentz 

Let us see in what way both the observers, outside 
and inside, describe what takes place in the lift. 

The outside observer notices the motion of the 
lift and of all bodies in the lift, and finds them 
in agreement with Newton's gravitational law. For 
him, the motion is not uniform, but accelerated, 
because of the action of the gravitational field of the 

However, a generation of physicists born and 
brought up in the lift would reason quite differently. 
They would believe themselves in possession of an 
inertial system and would refer all laws of nature to 
their lift, stating with justification that the laws take on 
a specially simple form in their c.s. It would be natural 
for them to assume their lift at rest and their c.s. the 
inertial one. 

It is impossible to settle the differences between the 
outside and the inside observers. Each of them could 
claim the right to refer all events to his c.s. Both de- 
scriptions of events could be made equally consistent. 

We see from this example that a consistent description 
of physical phenomena in two different c.s. is possible, 
even if they are not moving uniformly, relative to each 
other. But for such a description we must take into 
account gravitation, building, so to speak, the "bridge" 
which effects a transition from one c.s. to the other. 


The gravitational field exists for the outside observer; 
it does not for the inside observer. Accelerated motion 
of the lift in the gravitational field exists for the outside 
observer, rest and absence of the gravitational field for 
the inside observer. But the "bridge", the gravitational 
field, making the description in both c.s. possible, rests 
on one very important pillar : the equivalence of gravi- 
tational and inertial mass. Without this clue, unnoticed 
in classical mechanics, our present argument would fail 

Now for a somewhat different idealized experiment. 
There is, let us assume, an inertial c.s., in which the 
law of inertia is valid. We have already described 


what happens in a lift resting in such an inertial c.s. 
But we now change our picture. Someone outside 
has fastened a rope to the lift and is pulling, with a 
constant force, in the direction indicated in our draw- 


ing. It is immaterial how this is done. Since the laws 
of mechanics are valid in this c.s., the whole lift moves 
with a constant acceleration in the direction of the 
motion. Again we shall listen to the explanation of 
phenomena going on in the lift and given by both the 
outside and inside observers. 

The outside observer: My c.s. is an inertial one. The 
lift moves with constant acceleration, because a 
constant force is acting. The observers inside are 
in absolute motion, for them the laws of mechanics 
are invalid. They do not find that bodies, on which 
no forces are acting, are at rest. If a body is left free, 
it soon collides with the floor of the lift, since the 
floor moves upward toward the body. This happens 
exactly in the same way for a watch and for a hand- 
kerchief. It seems very strange to me that the ob- 
server inside the lift must always be on the "floor", 
because as soon as he jumps the floor will reach him 

The inside observer: I do not see any reason for believing 
that my lift is in absolute motion. I agree that my c.s., 
rigidly connected with my lift, is not really inertial, but 
I do not believe that it has anything to do with absolute 
motion. My watch, my handkerchief, and all bodies 
are falling because the whole lift is in a gravitational 
field. I notice exactly the same kinds of motion as the 
man on the earth. He explains them very simply by 
the action of a gravitational field. The same holds good 
for me. 

These two descriptions, one by the outside, the other 


by the inside, observer, are quite consistent, and there 
is no possibility of deciding which of them is right. We 
may assume either one of them for the description of 
phenomena in the lift: either non-uniform motion and 
absence of a gravitational field with the outside observer, 
or rest and the presence of a gravitational field with the 
inside observer. 

The outside observer may assume that the lift 
is in "absolute" non-uniform motion. But a motion 
which is wiped out by the assumption of an acting 
gravitational field cannot be regarded as absolute 

There is, possibly, a way out of the ambiguity of two 
such different descriptions, and a decision in favour of 
one against the other could perhaps be made. Imagine 
that a light ray enters the lift horizontally through a side 
window and reaches the opposite wall after a very short 
time. Again let us see how the path of the light would 
be predicted by the two observers. 

The outside observer, believing in accelerated motion 
of the lift, would argue: The light ray enters the 
window and moves horizontally, along a straight 
line and with a constant velocity, toward the opposite 
wall. But the lift moves upward, and during the 
time in which the light travels toward the wall the 
lift changes its position. Therefore, the ray will meet 
at a point not exactly opposite its point of entrance, but 
a little below. The difference will be very slight, but 
it exists nevertheless, and the light ray travels, relative 
to the lift, not along a straight, but along a slightly 


curved line. The difference is due to the distance 
covered by the lift during the time the ray is crossing 
the interior. 

The inside observer, who believes in the gravitational 
field acting on all objects in his lift, would say: there 
is no accelerated motion of the lift, but only the action 
of the gravitational field. A beam of light is weightless 
and, therefore, will not be affected by the gravitational 
field. If sent in a horizontal direction, it will meet 
the wall at a point exactly opposite to that at which it 

It seems from this discussion that there is a possibility 
of deciding between these two opposite points of view 
as the phenomenon would be different for the two 
observers. If there is nothing illogical in either of the 
explanations just quoted, then our whole previous 
argument is destroyed, and we cannot describe all 
phenomena in two consistent ways, with and without 
a gravitational field. 


But there is, fortunately, a grave fault in the reasoning 
of the inside observer, which saves our previous con- 
clusion. He said: "A beam of light is weightless and, 
therefore, will not be affected by the gravitational 
field." This cannot be right ! A beam of light carries 
energy and energy has mass. But every inertial mass 
is attracted by the gravitational field, as inertial and 
gravitational masses are equivalent. A beam of light 
will bend in a gravitational field exactly as a body 
would if thrown horizontally with a velocity equal to that 
of light. If the inside observer had reasoned correctly 
and had taken into account the bending of light rays in 
a gravitational field, then his results would have been 
exactly the same as those of an outside observer. 

The gravitational field of the earth is, of course, too 
weak for the bending of light rays in it to be proved 
directly, by experiment. But the famous experiments 
performed during the solar eclipses show, conclusively 
though indirectly, the influence of a gravitational field 
on the path of a light ray. 

It follows from these examples that there is a well- 
founded hope of formulating a relativistic physics. But 
for this we must first tackle the problem of gravitation. 

We saw from the example of the lift the consistency 
of the two descriptions. Non-uniform motion may, 
or may not, be assumed. We can eliminate "absolute" 
motion from our examples by a gravitational field. 
But then there is nothing absolute in the non-uniform 
motion. The gravitational field is able to wipe it out 


The ghosts of absolute motion and inertial c.s. can 
be expelled from physics and a new relativistic physics 
built. Our idealized experiments show how the problem 
of the general relativity theory is closely connected 
with that of gravitation and why the equivalence 
of gravitational and inertial mass is so essential for 
this connection. It is clear that the solution of the 
gravitational problem in the general theory of relativity 
must differ from the Newtonian one. The laws of 
gravitation must, just as all laws of nature, be formu- 
lated for all possible c.s., whereas the laws of classical 
mechanics, as formulated by Newton, are valid only in 
inertial c.s. 


Our next example will be even more fantastic than 
the one with the falling lift. We have to approach 
a new problem; that of a connection between the 
general relativity theory and geometry. Let us begin 
with the description of a world in which only two- 
dimensional and not, as in ours, three-dimensional 
creatures live. The cinema has accustomed us to two- 
dimensional creatures acting on a two-dimensional 
screen. Now let us imagine that these shadow figures, 
that is, the actors on the screen, really do exist, that 
they have the power of thought, that they can create 
their own science, that for them a two-dimensional 
screen stands for geometrical space. These creatures are 
unable to imagine, in a concrete way, a three-dimen- 
sional space just as we are unable to imagine a world of 


four dimensions. They can deflect a straight line; they 
know what a circle is, but they are unable to construct 
a sphere, because this would mean forsaking their two- 
dimensional screen. We are in a similar position. We 
are able to deflect and curve lines and surfaces, but we 
can scarcely picture a deflected and curved three- 
dimensional space. 

By living, thinking, and experimenting, our shadow 
figures could eventually master the knowledge of the 
two-dimensional Euclidean geometry. Thus, they could 
prove, for example, that the sum of the angles in a 
triangle is 180 degrees. They could construct two circles 
with a common centre, one very small, the other large. 
They would find that the ratio of the circumferences of 
two such circles is equal to the ratio of their radii, a 
result again characteristic of Euclidean geometry. If 
the screen were infinitely great, these shadow beings 
would find that once having started a journey straight 
ahead, they would never return to their point of 

Let us now imagine these two-dimensional creatures 
living in changed conditions. Let us imagine that some- 
one from the outside, the "third dimension", transfers 
them from the screen to the surface of a sphere with 
a very great radius. If these shadows are very small in 
relation to the whole surface, if they have no means of 
distant communication and cannot move very far, then 
they will not be aware of any change. The sum of angles 
in small triangles still amounts to 180 degrees. Two 
small circles with a common centre still show that the 


ratio of their radii and circumferences are equal. A 
journey along a straight line never leads them back to 
the starting-point. 

But let these shadow beings, in the course of time, 
develop their theoretical and technical knowledge. Let 
them find means of communication which will enable 
them to cover large distances swiftly. They will then 
find that starting on a journey straight ahead, they 
ultimately return to their point of departure. " Straight 
ahead" means along the great circle of the sphere. 
They will also find that the ratio of two circles with a 
common centre is not equal to the ratio of the radii, if 
one of the radii is small and the other great. 

If our two-dimensional creatures are conservative, if 
they have learned the Euclidean geometry for genera- 
tions past when they could not travel far and when 
this geometry fitted the facts observed, they will 
certainly make every possible effort to hold on to it, 
despite the evidence of their measurements. They could 
try to make physics bear the burden of these discrep- 
ancies. They could seek some physical reasons, say 
temperature differences, deforming the lines and causing 
deviation from Euclidean geometry. But, sooner or 
later, they must find out that there is a much more 
logical and convincing way of describing these occur- 
rences. They will eventually understand that their 
world is a finite one, with different geometrical prin- 
ciples from those they learned. They will understand 
that, in spite of their inability to imagine it, their 
world is the two-dimensional surface of a sphere. They 


will soon learn new principles of geometry, which 
though differing from the Euclidean can, nevertheless, 
be formulated in an equally consistent and logical way 
for their two-dimensional world. For the new generation 
brought up with a knowledge of the geometry of the 
sphere, the old Euclidean geometry will seem more 
complicated and artificial since it does not fit the facts 

Let us return to the three-dimensional creatures of 
our world. 

What is meant by the statement that our three- 
dimensional space has a Euclidean character? The 
meaning is that all logically proved statements of the 
Euclidean geometry can also be confirmed by actual 
experiment. We can, with the help of rigid bodies or 
light rays, construct objects corresponding to the 
idealized objects of Euclidean geometry. The edge of 
a ruler or a light ray corresponds to the line; the sum 
of the angles of a triangle built of thin rigid rods is 180 
degrees; the ratio of the radii of two circles with a 
common centre constructed from thin unbendable wire 
is equal to that of their circumference. Interpreted in 
this way, the Euclidean geometry becomes a chapter 
of physics, though a very simple one. 

But we can imagine that discrepancies have been 
discovered: for instance, that the sum of the angles of 
a large triangle constructed from rods, which for many 
reasons had to be regarded as rigid, is not 180 degrees. 
Since we are already used to the idea of the concrete 
representation of the objects of Euclidean geometry 


by rigid bodies, we should probably seek some physical 
force as the cause of such unexpected misbehaviour of 
our rods. We should try to find the physical nature of 
this force and its influence on other phenomena. To 
save the Euclidean geometry, we should accuse the 
objects of not being rigid, of not exactly corresponding 
to those of Euclidean geometry. We should try to find 
a better representation of bodies behaving in the way 
expected by Euclidean geometry. If, however, we 
should not succeed in combining Euclidean geometry 
and physics into a simple and consistent picture, we 
should have to give up the idea of our space being 
Euclidean and seek a more convincing picture of reality 
under more general assumptions about the geometrical 
character of our space. 

The necessity for this can be illustrated by an idealized 
experiment showing that a really relativistic physics 
cannot be based upon Euclidean geometry. Our argu- 
ment will imply results already learned about inertial 
c.s. and the special relativity theory. 

Imagine a large disc with two circles with a common 
centre drawn on it, one very small, the other very large. 
The disc rotates quickly. The disc is rotating relative to 
an outside observer, and there is an inside observer on 
the disc. We further assume that the c.s. of the outside 
observer is an inertial one. The outside observer may 
draw, in his inertial c.s., the same two circles, small and 
large, resting in his c.s. but coinciding with the circles 
on the rotating disc. Euclidean geometry is valid in his 
c.s. since it is inertial, so that he will find the ratio of 


the circumferences equal to that of the radii. But how 
about the observer on the disc? From the point of view 
of classical physics and also the special relativity theory, 
his c.s. is a forbidden one. But if we intend to find new 
forms for physical laws, valid in any c.s., then we must 
treat the observer on the disc and the observer outside 

with equal seriousness. We, from the outside, are now 
watching the inside observer in his attempt to find, by 
measurement, the circumferences and radii on the 
rotating disc. He uses the same small measuring stick 
used by the outside observer. "The same" means either 
really the same, that is, handed by the outside observer 
to the inside, or, one of two sticks having the same length 
when at rest in a c.s. 

The inside observer on the disc begins measuring the 


radius and circumference of the small circle. His result 
must be the same as that of the outside observer. The 
axis on which the disc rotates passes through the centre. 
Those parts of the disc near the centre have very 
small velocities. If the circle is small enough, we can 
safely apply classical mechanics and ignore the special 
relativity theory. This means that the stick has the same 
length for the outside and inside observers, and the 
result of these two measurements will be the same for 
them both. Now the observer on the disc measures the 
radius of the large circle. Placed on the radius, the 
stick moves, for the outside observer. Such a stick, 
however, does not contract and will have the same 
length for both observers, since the direction of the 
motion is perpendicular to the stick. Thus three mea- 
surements are the same for both observers: two radii 
and the small circumference. But it is not so with the 
fourth measurement ! The length of the large circum- 
ference will be different for the two observers. The 
stick placed on the circumference in the direction of 
the motion will now appear contracted to the outside 
observer, compared to his resting stick. The velocity is 
much greater than that of the inner circle, and this 
contraction should be taken into account. If, therefore, 
we apply the results of the special relativity theory, 
our conclusion here is : the length of the great circum- 
ference must be different if measured by the two 
observers. Since only one of the four lengths measured 
by the two observers is not the same for them both, 
the ratio of the two radii cannot be equal to the ratio of 


the two circumferences for the inside observer as it is 
for the outside one. This means that the observer on the 
disc cannot confirm the validity of Euclidean geometry 
in his c.s. 

After obtaining this result, the observer on the disc 
could say that he does not wish to consider c.s. in which 
Euclidean geometry is not valid. The breakdown of the 
Euclidean geometry is due to absolute rotation, to the 
fact that his c.s. is a bad and forbidden one. But, in 
arguing in this way, he rejects the principal idea of the 
general theory of relativity. On the other hand, if we 
wish to reject absolute motion and to keep up the idea 
of the general theory of relativity, then physics must all 
be built on the basis of a geometry more general than 
the Euclidean. There is no way of escape from this 
consequence if all c.s. are permissible. 

The changes brought about by the general relativity 
theory cannot be confined to space alone. In the special 
relativity theory we had clocks resting in every c.s., 
having the same rhythm and synchronized, that is, 
showing the same time simultaneously. What happens 
to a clock in a non-inertial c.s.? The idealized experi- 
ment with the disc will again be of use. The outside 
observer has in his inertial c.s. perfect clocks all having 
the same rhythm, all synchronized. The inside observer 
takes two clocks of the same kind and places one on 
the small inner circle and the other on the large outer 
circle. The clock on the inner circle has a very small 
velocity relative to the outside observer. We can, 
therefore, safely conclude that its rhythm will be the 


same as that of the outside clock. But the clock on the 
large circle has a considerable velocity, changing its 
rhythm compared to the clocks of the outside observer 
and, therefore, also compared to the clock placed on 
the small circle. Thus, the two rotating clocks will have 
different rhythms and, applying the results of the special 
relativity theory, we again see that in our rotating c.s. 
we can make no arrangements similar to those in an 
inertial c.s. 

To make clear what conclusions can be drawn from 
this and previously described idealized experiments, let 
us once more quote a dialogue between the old physicist 
0, who believes in classical physics, and the modern 
physicist M, who knows the general relativity theory. 
is the outside observer, in the inertial c.s., whereas 
M is on the rotating disc. 

0. In your c.s., Euclidean geometry is not valid. I 
watched your measurements and I agree that the ratio 
of the two circumferences is not, in your c.s., equal to 
the ratio of the two radii. But this shows that your c.s. 
is a forbidden one. My c.s., however, is of an inertial 
character, and I can safely apply Euclidean geometry. 
Your disc is in absolute motion and, from the point of 
view of classical physics, forms a forbidden c.s., in 
which the laws of mechanics are not valid. 

M. I do not want to hear anything about absolute 
motion. My c.s. is just as good as yours. What I noticed 
was your rotation relative to my disc. No one can forbid 
me to relate all motions to my disc. 

0. But did you not feel a strange force trying to 


keep you away from the centre of the disc? If your 
disc were not a rapidly rotating merry-go-round, the 
two things which you observed would certainly not 
have happened. You would not have noticed the force 
pushing you toward the outside nor would you have 
noticed that Euclidean geometry is not applicable in 
your c.s. Are not these facts sufficient to convince you 
that your c.s. is in absolute motion? 

M. Not at all ! I certainly noticed the two facts you 
mention, but I hold a strange gravitational field acting 
on my disc responsible for them both. The gravita- 
tional field, being directed toward the outside of the 
disc, deforms my rigid rods and changes the rhythm 
of my clocks. The gravitational field, non-Euclidean 
geometry, clocks with different rhythms are, for me, all 
closely connected. Accepting any c.s., I must at the 
same time assume the existence of an appropriate 
gravitational field with its influence upon rigid rods 
and clocks. 

0. But are you aware of the difficulties caused by 
your general relativity theory? I should like to make 
my point clear by taking a simple non-physical example. 
Imagine an idealized American town consisting of 
parallel streets with parallel avenues running perpen- 
dicular to them. The distance between the streets and 
also between the avenues is always the same. With 
these assumptions fulfilled, the blocks are of exactly 
the same size. In this way I can easily characterize the 
position of any block. But such a construction would 
be impossible without Euclidean geometry. Thus, for 


instance, we cannot cover our whole earth with one 
great ideal American town. One look at the globe will 
convince you. But neither could we cover your disc 
with such an " American town construction". You claim 
that your rods are deformed by the gravitational field. 
The fact that you could not confirm Euclid's theorem 
about the equality of the ratio of radii and circumfer- 
ences shows clearly that if you carry such a construction 
of streets and avenues far enough you will, sooner or 
later, get into difficulties and find that it is impossible 
on your disc. Your geometry on your rotating disc 
resembles that on a curved surface, where, of course, 
the streets-and-avenues construction is impossible on a 
great enough part of the surface. For a more physical 
example take a plane irregularly heated with different 
temperatures on different parts of the surface. Can 
you, with small iron sticks expanding in length with 
temperature, carry out the " parallel-perpendicular" 
construction which I have drawn below? Of course 
not! Your "gravitational field" plays the same tricks 


on your rods as the change of temperature on the small 
iron sticks. 

M. All this does not frighten me. The street-avenue 
construction is needed to determine positions of points, 
with the clock to order events. The town need not be 
American, it could just as well be ancient European. 
Imagine your idealized town made of plasticine and 
then deformed. I can still number the blocks and recog- 
nize the streets and avenues, though these are no longer 

straight and equidistant. Similarly, on our earth, 
longitude and latitude denote the positions of points, 
although there is no "American town" construction. 

0. But I still see a difficulty. You are forced to use 
your "European town structure' 5 . I agree that you can 
order points, or events^ but this construction will muddle 
all measurement of distances. It will not give you the 
metric properties of space as does my construction. Take 
an example. I know, in my American town, that to 
walk ten blocks I have to cover twice the distance of 


five blocks. Since I know that all blocks are equal, I 
can immediately determine distances. 

M. That is true. In my " European town" structure, 
I cannot measure distances immediately by the number 
of deformed blocks. I must know something more; I 
must know the geometrical properties of my surface. 
Just as everyone knows that from o to 10 longitude 
on the Equator is not the same distance as from o to 
10 longitude near the North Pole. But every navigator 
knows how to judge the distance between two such 
points on our earth because he knows its geometrical 
properties. He can either do it by calculations based on 
the knowledge of spherical trigonometry, or he can do 
it experimentally, sailing his ship through the two 
distances at the same speed. In your case the whole 
problem is trivial, because all the streets and avenues are 
the same distance apart. In the case of our earth it is 
more complicated; the two meridians o and 10 meet 
at the earth's poles and are farthest apart on the 
Equator. Similarly, in my "European town structure", 
I must know something more than you in your "Ameri- 
can town structure", in order to determine distances. 
I can gain this additional knowledge by studying the 
geometrical properties of my continuum in every 
particular case. 

0. But all this only goes to show how inconvenient 
and complicated it is to give up the simple structure of 
the Euclidean geometry for the intricate scaffolding 
which you are bound to use. Is this really necessary? 

M . I am afraid it is, if we want to apply our physics 


to any c.s., without the mysterious inertial c.s. I agree 
that my mathematical tool is more complicated than 
yours, but my physical assumptions are simpler and 
more natural. 

The discussion has been restricted to two-dimen- 
sional continua. The point at issue in the general 
relativity theory is still more complicated, since it is not 
the two-dimensional, but the four-dimensional time- 
space continuum. But the ideas are the same as those 
sketched in the two-dimensional case. We cannot use in 
the general relativity theory the mechanical scaffolding 
of parallel, perpendicular rods and synchronized clocks, 
as in the special relativity theory. In an arbitrary c.s. 
we cannot determine the point and the instant at which 
an event happens by the use of rigid rods, rhythmical 
and synchronized clocks, as in the inertial c.s. of the 
special relativity theory. We can still order the events 
with our non-Euclidean rods and our clocks out of 
rhythm. But actual measurements requiring rigid rods 
and perfect rhythmical and synchronized clocks can be 
performed only in the local inertial c.s. For this the 
whole special relativity theory is valid; but our "good" 
c.s. is only local, its inertial character being limited in 
space and time. Even in our arbitrary c.s. we can 
foresee the results of measurements made in the local 
inertial c.s. But for this we must know the geometrical 
character of our time-space continuum. 

Our idealized experiments indicate only the general 
character of the new relativistic physics. They show us 
that our fundamental problem is that of gravitation. 


They also show us that the general relativity theory 
leads to further generalization of time and space 

The general theory of relativity attempts to formulate 
physical laws for all c.s. The fundamental problem 
of the theory is that of gravitation. The theory makes 
the first serious effort, since Newton's time, to reformu- 
late the law of gravitation. Is this really necessary? 
We have already learned about the achievements of 
Newton's theory, about the great development of 
astronomy based upon his gravitational law. Newton's 
law still remains the basis of all astronomical calcula- 
tions. But we also learned about some objections to the 
old theory. Newton's law is valid only in the inertia! 
c.s. of classical physics, in c.s. defined, we remember, 
by the condition that the laws of mechanics must be 
valid in them. The force between two masses depends 
upon their distance from each other. The connection 
between force and distance is, as we know, invariant 
with respect to the classical transformation. But this 
law does not fit the frame of special relativity. The 
distance is not invariant with respect to the Lorentz 
transformation. We could try, as we did so successfully 
with the laws of motion, to generalize the gravitational 
law, to make it fit the special relativity theory, or, in 
other words, to formulate it so that it would be invariant 
with respect to the Lorentz and not to the classical 
transformation. But Newton's gravitational law op- 


posed obstinately all our efforts to simplify and fit it 
into the scheme of the special relativity theory. Even 
if we succeeded in this, a further step would still be 
necessary: the step from the inertial c.s. of the special 
relativity theory to the arbitrary c.s. of the general 
relativity theory. On the other hand, the idealized 
experiments about the falling lift show clearly that 
there is no chance of formulating the general relativity 
theory without solving the problem of gravitation. 
From our argument we see why the solution of the 
gravitational problem will differ in classical physics and 
general relativity. 

We have tried to indicate the way leading to the 
general relativity theory and the reasons forcing us to 
change our old views once more. Without going into 
the formal structure of the theory, we shall characterize 
some features of the new gravitational theory as com- 
pared with the old. It should not be too difficult to 
grasp the nature of these differences in view of all that 
has previously been said. 

(1) The gravitational equations of the general rela- 
tivity theory can be applied to any c.s. It is merely a 
matter of convenience to choose any particular c.s. in 
a special case. Theoretically all c.s. are permissible. By 
ignoring the gravitation, we automatically come back 
to the inertial c.s. of the special relativity theory. 

(2) Newton's gravitational law connects the motion 
of a body here and now with the action of a body at 
the same time in the far distance. This is the law which 
formed a pattern for our whole mechanical view. But 


the mechanical view broke down. In Maxwell's equa- 
tions we realized a new pattern for the laws of nature. 
Maxwell's equations are structure laws. They connect 
events which happen now and here with events which 
will happen a little later in the immediate vicinity. 
They are the laws describing the changes of the electro- 
magnetic field. Our new gravitational equations are 
also structure laws describing the changes of the gravi- 
tational field. Schematically speaking, we could say: 
the transition from Newton's gravitational law to 
general relativity resembles somewhat the transition 
from the theory of electric fluids with Coulomb's law 
to Maxwell's theory. 

(3) Our world is not Euclidean. The geometrical 
nature of our world is shaped by masses and their 
velocities. The gravitational equations of the general 
relativity theory try to disclose the geometrical pro- 
perties of our world. 

Let us suppose, for the moment, that we have suc- 
ceeded in carrying out consistently the programme of 
the general relativity theory. But are we not in danger 
of carrying speculation too far from reality? We know 
how well the old theory explains astronomical observa- 
tions. Is there a possibility of constructing a bridge 
between the new theory and observation? Every specu- 
lation must be tested by experiment, and any results, 
no matter how attractive, must be rejected if they do 
not fit the facts. How did the new theory of gravita- 
tion stand the test of experiment? This question can be 
answered in one sentence: The old theory is a special 


limiting case of the new one. If the gravitational forces 
are comparatively weak, the old Newtonian law turns 
out to be a good approximation to the new laws of 
gravitation. Thus all observations which support the 
classical theory also support the general relativity 
theory. We regain the old theory from the higher level 
of the new one. 

Even if no additional observation could be quoted in 
favour of the new theory, if its explanation were only 
just as good as the old one, given a free choice between 
the two theories, we should have to decide in favour 
of the new one. The equations of the new theory are, 
from the formal point of view, more complicated, but 
their assumptions are, from the point of view of funda- 
mental principles, much simpler. The two frightening 
ghosts, absolute time and an inertial system, have 
disappeared. The clue of the equivalence of gravita- 
tional and inertial mass is not overlooked. No assump- 
tion about the gravitational forces and their dependence 
on distance is needed. The gravitational equations have 
the form of structure laws, the form required of all 
physical laws since the great achievements of the field 

Some new deductions, not contained in Newton's 
gravitational law, can be drawn from the new gravita- 
tional laws. One, the bending of light rays in a gravi- 
tational field, has already been quoted. Two further 
consequences will now be mentioned. 

If the old laws follow from the new one when the 
gravitational forces are weak, the deviations from the 


Newtonian law of gravitation can be expected only for 
comparatively strong gravitational forces. Take our 
solar system. The planets, our earth among them, move 
along elliptical paths around the sun. Mercury is the 
planet nearest the sun. The attraction between the sun 
and Mercury is stronger than that between the sun and 
any other planet, since the distance is smaller. If there 
is any hope of finding a deviation from Newton's law, 
the greatest chance is in the case of Mercury. It 
follows, from classical theory, that the path described 
by Mercury is of the same kind as that of any other 
planet except that it is nearer the sun. According to 

the general relativity theory, the motion should be 
slightly different. Not only should Mercury travel 
around the sun, but the ellipse which it describes should 
rotate very slowly, relative to the c.s. connected with 
the sun. This rotation of the ellipse expresses the new 
effect of the general relativity theory. The new theory 


predicts the magnitude of this effect. Mercury's ellipse 
would perform a complete rotation in three million 
years ! We see how small the effect is, and how hopeless 
it would be to seek it in the case of planets farther 
removed from the sun. 

The deviation of the motion of the planet Mercury 
from the ellipse was known before the general relativity 
theory was formulated, and no explanation could be 
found. On the other hand, general relativity developed 
without any attention to this special problem. Only later 
was the conclusion about the rotation of the ellipse in the 
motion of a planet around the sun drawn from the new 
gravitational equations. In the case of Mercury, theory 
explained successfully the deviation of the motion from 
the Newtonian law. 

But there is still another conclusion which was drawn 
from the general relativity theory and compared with 
experiment. We have already seen that a clock placed 
on the large circle of a rotating disc has a different 
rhythm from one placed on the smaller circle. Similarly, 
it follows from the theory of relativity that a clock 
placed on the sun would have a different rhythm from 
one placed on the earth, since the influence of the 
gravitational field is much stronger on the sun than on 
the earth. 

We remarked on p. 103 that sodium, when incan- 
descent, emits homogeneous yellow light of a definite 
wave-length. In this radiation the atom reveals one of 
its rhythms; the atom represents, so to speak, a clock 
and the emitted wave-length one of its rhythms. Accord- 


ing to the general theory of relativity, the wave-length 
of light emitted by a sodium atom, say, placed on the 
sun should be very slightly greater than that of light 
emitted by a sodium atom on our earth. 

The problem of testing the consequences of the 
general relativity theory by observation is an intricate 
one and by no means definitely settled. As we are 
concerned with principal ideas, we do not intend to go 
deeper into this matter, and only state that the verdict 
of experiment seems, so far, to confirm the conclusions 
drawn from the general relativity theory. 


We have seen how and why the mechanical point of 
view broke down. It was impossible to explain all 
phenomena by assuming that simple forces act be- 
tween unalterable particles. Our first attempts to go 
beyond the mechanical view and to introduce field 
concepts proved most successful in the domain of elec- 
tromagnetic phenomena. The structure laws for the 
electromagnetic field were formulated; laws connect- 
ing events very near to each other in space and time. 
These laws fit the frame of the special relativity theory, 
since they are invariant with respect to the Lorentz 
transformation. Later the general relativity theory 
formulated the gravitational laws. Again they are 
structure laws describing the gravitational field between 
material particles. It was also easy to generalize Max- 
well's laws so that they could be applied to any G.S., like 
the gravitational laws of the general relativity theory. 


We have two realities: matter and field. There is no 
doubt that we cannot at present imagine the whole of 
physics built upon the concept of matter as the physi- 
cists of the early nineteenth century did. For the 
moment we accept both the concepts. Can we think of 
matter and field as two distinct and different realities? 
Given a small particle of matter, we could picture in a 
naive way that there is a definite surface of the particle 
where it ceases to exist and its gravitational field ap- 
pears. In our picture, the region in which the laws of 
field are valid is abruptly separated from the region 
in which matter is present. But what are the physical 
criterions distinguishing matter and field? Before we 
learned about the relativity theory we could have tried 
to answer this question in the following way: matter 
has mass, whereas field has not. Field represents 
energy, matter represents mass. But we already know 
that such an answer is insufficient in view of the further 
knowledge gained. From the relativity theory we know 
that matter represents vast stores of energy and that 
energy represents matter. We cannot, in this way, 
distinguish qualitatively between matter and field, 
since the distinction between mass and energy is not 
a qualitative one. By far the greatest part of energy is 
concentrated in matter; but the field surrounding the 
particle also represents energy, though in an incom- 
parably smaller quantity. We could therefore say: 
Matter is where the concentration of energy is great, 
field where the concentration of energy is small. But 
if this is the case, then the difference between matter 


and field is a quantitative rather than a qualitative one. 
There is no sense in regarding matter and field as two 
qualities quite different from each other. We cannot 
imagine a definite surface separating distinctly field and 

The same difficulty arises for the charge and its 
field. It seems impossible to give an obvious qualitative 
criterion for distinguishing between matter and field or 
charge and field. 

Our structure laws, that is, Maxwell's laws and the 
gravitational laws, break down for very great concen- 
trations of energy or, as we may say, where sources of 
the field, that is electric charges or matter, are present. 
But could we not slightly modify our equations so that 
they would be valid everywhere, even in regions where 
energy is enormously concentrated? 

We cannot build physics on the basis of the matter- 
concept alone. But the division into matter and field 
is, after the recognition of the equivalence of mass and 
energy, something artificial and not clearly defined. 
Could we not reject the concept of matter and build 
a pure field physics? What impresses our senses as 
matter is really a great concentration of energy into a 
comparatively small space. We could regard matter as 
the regions in space where the field is extremely strong. 
In this way a new philosophical background could be 
created. Its final aim would be the explanation of all 
events in nature by structure laws valid always and 
everywhere. A thrown stone is, from this point of 
view, a changing field, where the states of greatest field 


intensity travel through space with the velocity of the 
stone. There would be no place, in our new physics, 
for both field and matter, field being the only reality. 
This new view is suggested by the great achievements 
of field physics, by our success in expressing the laws 
of electricity, magnetism, gravitation in the form of 
structure laws, and finally by the equivalence of mass 
and energy. Our ultimate problem would be to modify 
our field laws in such a way that they would not break 
down for regions in which the energy is enormously 

But we have not so far succeeded in fulfilling this 
programme convincingly and consistently. The decision, 
as to whether it is possible to carry it out, belongs to 
the future. At present we must still assume in all our 
actual theoretical constructions two realities: field and 

Fundamental problems are still before us. We know 
that all matter is constructed from a few kinds of 
particles only. How are the various forms of matter 
built from these elementary particles? How do these 
elementary particles interact with the field? By the 
search for an answer to these questions new ideas have 
been introduced into physics, the ideas of the quantum 


A new concept appears in physics, the' most important 
invention since Newton's time: the field. It needed great 
scientific imagination to realize that it is not the charges 


nor the particles but the field in the space between the 
charges and the particles which is essential for the descrip- 
tion of physical phenomena. The field concept proves most 
successful and leads to the formulation of MaxweWs equa- 
tions describing the structure of the electromagnetic field 
and governing the electric as well as the optical pheno- 

The theory of relativity arises from the field problems. The 
contradictions and inconsistencies of the old theories force us to 
ascribe new properties to the time-space continuum, to the scene 
of all events in our physical world. 

The relativity theory develops in two steps. The first 
step leads to what is known as the special theory of rela- 
tivity, applied only to inertial co-ordinate systems, that is, 
to systems in which the law of inertia, as formulated by 
Newton, is valid. The special theory of relativity is based 
on two fundamental assumptions: physical laws are the 
same in all co-ordinate systems moving uniformly, rela- 
tive to each other; the velocity of light always has the same 
value. From these assumptions, fully confirmed by experi- 
ment, the properties of moving rods and clocks, their changes 
in length and rhythm depending on velocity, are deduced. 
The theory of relativity changes the laws of mechanics. The 
old laws are invalid if the velocity of the moving particle 
approaches that of light. The new laws for a moving body 
as reformulated by the relativity theory are splendidly con- 
firmed by experiment. A further consequence of the (special) 
theory of relativity is the connection between mass and 
energy. Mass is energy and energy has mass. The two 
conservation laws of mass and energy are combined by the 



relativity theory into one, the conservation law of mass- 

The general theory of relativity gives a still deeper ana- 
lysis of the time-space continuum. The validity of the theory 
is no longer restricted to inertial co-ordinate systems. 
The theory attacks the problem of gravitation and for- 
mulates new structure laws for the gravitational field. 
It forces us to analyse the role played by geometry in the 
description of the physical world. It regards the fact that 
gravitational and inertial mass are equal, as essential and 
not merely accidental, as in classical mechanics. The ex- 
perimental consequences of the general relativity theory 
differ only slightly from those of classical mechanics. Thy 
stand the test of experiment well wherever comparison is 
possible. But the strength of the theory lies in its inner 
consistency and the simplicity of its fundamental assump- 

The theory of relativity stresses the importance of the field 
concept in physics. But we have not yet succeeded in formulating 
a pure field physics. For the present we must still assume the 
existence of both: field and matter. 



Continuity, discontinuity Elementary quanta of matter and 
electricity The quanta of light Light spectra The waves 
of matter Probability waves Physics and reality 


A MAP of New York City and the surrounding country 
is spread before us. We ask: which points on this map 
can be reached by train? After looking up these points 
in a railway timetable, we mark them on the map. 
We now change our question and ask: which points 
can be reached by car? If we draw lines on the map 
representing all the roads starting from New York, every 
point on these roads can, in fact, be reached by car. 
In both cases we have sets of points. In the first they 
are separated from each other and represent the different 
railway stations, and in the second they are the points 
along the lines representing the roads. Our next 
question is about the distance of each of these points 
from New York, or, to be more rigorous, from a certain 
spot in that city. In the first case, certain numbers 
correspond to the points on our map. These numbers 
change by irregular, but always finite, leaps and bounds. 
We say: the distances from New York of the places 
which can be reached by train change only in a 
discontinuous way. Those of the places which can be 



reached by car, however, may change by steps as small 
as we wish, they can vary in a continuous way. The changes 
in distance can be made arbitrarily small in the case of 
a car, but not in the case of a train. 

The output of a coal mine can change in a continu- 
ous way. The amount of coal produced can be de- 
creased or increased by arbitrarily small steps. But the 
number of miners employed can change only discon- 
tinuously. It would be pure nonsense to say: "Since 
yesterday, the number of employees has increased by 


Asked about the amount of money in his pocket, a 
man can give a number containing only two decimals. 
A sum of money can change only by jumps, in a dis- 
continuous way. In America the smallest permissible 
change or, as we shall call it, the " elementary quantum ' ' 
for American money, is one cent. The elementary 
quantum for English money is one farthing, worth 
only half the American elementary quantum. Here 
we have an example of two elementary quanta whose 
mutual values can be compared. The ratio of their 
values has a definite sense since one of them is worth 
twice as much as the other. 

We can say : some quantities can change continuously 
and others can change only discontinuously, by steps 
which cannot be further decreased. These indivisible 
steps are called the elementary quanta of the particular 
quantity to which they refer. 

We can weigh large quantities of sand and regard its 
mass as continuous even though its granular structure 


is evident. But if the sand were to become very precious 
and the scales used very sensitive, we should have to 
consider the fact that the mass always changes by a 
multiple number of one grain. The mass of this one 
grain would be our elementary quantum. From this 
example we see how the discontinuous character of a 
quantity, so far regarded as continuous, can be detected 
by increasing the precision of our measurements. 

If we had to characterize the principal idea of the 
quantum theory in one sentence, we could say: it must 
be assumed that some physical quantities so far regarded 
as continuous are composed of elementary quanta. 

The region of facts covered by the quantum theory is 
tremendously great. These facts have been disclosed by 
the highly developed technique of modern experiment. 
As we can neither show nor describe even the basic 
experiments, we shall frequently have to quote their 
results dogmatically. Our aim is to explain the prin- 
cipal underlying ideas only. 


In the picture of matter drawn by the kinetic theory, 
all elements are built of molecules. Take the simplest 
case of the lightest element, that is hydrogen. On p. 66 
we saw how the study of Brownian motions led to the 
determination of the mass of one hydrogen molecule. 
Its value is : 

0*000 ooo ooo ooo ooo ooo ooo 0033 gram. 


This means that mass is discontinuous. The mass of a 
portion of hydrogen can change only by a whole 
number of small steps each corresponding to the mass 
of one hydrogen molecule. But chemical processes show 
that the hydrogen molecule can be broken up into two 
parts, or, in other words, that the hydrogen molecule 
is composed of two atoms. In chemical processes it is 
the atom and not the molecule which plays the role of 
an elementary quantum. Dividing the above number 
by two, we find the mass of a hydrogen atom. This is 

0*000 ooo ooo ooo ooo ooo ooo 0017 gram. 

Mass is a discontinuous quantity. But, of course, we 
need not bother about this when determining weight. 
Even the most sensitive scales are far from attaining the 
degree of precision by which the discontinuity in mass 
variation could be detected. 

Let us return to a well-known fact. A wire is con- 
nected with the source of a current. The current is 
flowing through the wire from higher to lower poten- 
tial. We remember that many experimental facts were 
explained by the simple theory of electric fluids flowing 
through the wire. We also remember (p. 82) that the 
decision as to whether the positive fluid flows from 
higher to lower potential, or the negative fluid flows 
from lower to higher potential, was merely a matter of 
convention. For the moment we disregard all the 
further progress resulting from the field concepts. Even 
when thinking in the simple terms of electric fluids, 


there still remain some questions to be settled. As the 
name "fluid" suggests, electricity was regarded, in the 
early days, as a continuous quantity. The amount of 
charge could be changed, according to these old views, 
by arbitrarily small steps. There was no need to assume 
elementary electric quanta. The achievements of the 
kinetic theory of matter prepared us for a new question: 
do elementary quanta of electric fluids exist? The 
other question to be settled is : does the current consist 
of a flow of positive, negative or perhaps of both fluids? 
The idea of all the experiments answering these 
questions is to tear the electric fluid from the wire, to 
let it travel through empty space, to deprive it of any 
association with matter and then to investigate its 
properties, which must appear most clearly under these 
conditions. Many experiments of this kind were per- 
formed in the late nineteenth century. Before explaining 
the idea of these experimental arrangements, at least in 
one case, we shall quote the results. The electric fluid 
flowing through the wire is a negative one, directed, 
therefore, from lower to higher potential. Had we 
known this from the start, when the theory of electric 
fluids was first formed, we should certainly have inter- 
changed the words, and called the electricity of the 
rubber rod positive, that of the glass rod negative. It 
would then have been more convenient to regard the 
flowing fluid as the positive one. Since our first guess 
was wrong, we now have to put up with the incon- 
venience. The next important question is whether the 
structure of this negative fluid is "granular", whether 


or not it is composed of electric quanta. Again a number 
of independent experiments show that there is no doubt 
as to the existence of an elementary quantum of this 
negative electricity. The negative electric fluid is con- 
structed of grains, just as the beach is composed of 
grains of sand, or a house built of bricks. This result was 
formulated most clearly by J. J. Thomson, about forty 
years ago. The elementary quanta of negative electricity 
are called electrons. Thus every negative electric charge 
is composed of a multitude of elementary charges 
represented by electrons. The negative charge can, like 
mass, vary only discontinuously. The elementary electric 
charge is, however, so small that in many investigations 
it is equally possible and sometimes even more con- 
venient to regard it as a continuous quantity. Thus the 
atomic and electron theories introduce into science 
discontinuous physical quantities which can vary only 
by jumps. 

Imagine two parallel metal plates in some place from 
which all air has been extracted. One of the plates has 
a positive, the other a negative charge. A positive test 
charge brought between the two plates will be repelled 
by the positively charged and attracted by the nega- 
tively charged plate. Thus the lines of force of the 
electric field will be directed from the positively to the 
negatively charged plate. A force acting on a negatively 
charged test body would have the opposite direction. 
If the plates are sufficiently large, the lines of force 
between them will be equally dense everywhere; it is 
immaterial where the test body is placed, the force and, 


therefore, the density of the lines of force will be the 
same. Electrons brought somewhere between the plates 

would behave like raindrops in the gravitational field 
of the earth, moving parallel to each other from the 
negatively to the positively charged plate. There are 
many known experimental arrangements for bringing 
a shower of electrons into such a field which directs 
them all in the same way. One of the simplest is to 
bring a heated wire between the charged plates. Such 
a heated wire emits electrons which are afterwards 
directed by the lines of force of the external field. For 
instance, radio tubes, familiar to everyone, are based 
on this principle. 

Many very ingenious experiments have been per- 
formed on a beam of electrons. The changes of their 
path in different electric and magnetic external fields 
have been investigated. It has even been possible to 
isolate a single electron and to determine its elementary 
charge and its mass, that is, its inertial resistance to the 


action of an external force. Here we shall only quote 
the value of the mass of an electron. It turned out to 
be about two thousand times smaller than the mass of a 
hydrogen atom. Thus the mass of a hydrogen atom, 
small as it is, appears great in comparison with the mass 
of an electron. From the point of view of a consistent 
field theory, the whole mass, that is, the whole energy, 
of an electron is the energy of its field; the bulk of its 
strength is within a very small sphere, and away from 
the " centre" of the electron it is weak. 

We said before that the atom of any element is its 
smallest elementary quantum. This statement was be- 
lieved for a very long time. Now, however, it is no 
longer believed ! Science has formed a new view show- 
ing the limitations of the old one. There is scarcely any 
statement in physics more firmly founded on facts than 
the one about the complex structure of the atom. First 
came the realization that the electron, the elementary 
quantum of the negative electric fluid, is also one of the 
components of the atom, one of the elementary bricks 
from which all matter is built. The previously quoted 
example of a heated wire emitting electrons is only one 
of the numerous instances of the extraction of these 
particles from matter. This result closely connecting the 
problem of the structure of matter with that of electricity, 
follows, beyond any doubt, from very many independent 
experimental facts. 

It is comparatively easy to extract from an atom some 
of the electrons from which it is composed. This can be 
done by heat, as in our example of a heated wire, or in 


a different way, such as by bombarding atoms with other 

Suppose a thin, red-hot, metal wire is inserted into 
rarefied hydrogen. The wire will emit electrons in all 
directions. Under the action of a foreign electric field 
a given velocity will be imparted to them. An electron 
increases its velocity just like a stone falling in the 
gravitational field. By this method we can obtain a 
beam of electrons rushing along with a definite speed in 
a definite direction. Nowadays, we can reach velocities 
comparable to that of light by submitting electrons to 
the action of very strong fields. What happens, then, 
when a beam of electrons of a definite velocity impinges 
on the molecules of rarefied hydrogen? The impact of a 
sufficiently speedy electron will not only disrupt the 
hydrogen molecule into its two atoms but will also 
extract an electron from one of the atoms. 

Let us accept the fact that electrons are constituents 
of matter. Then, an atom from which an electron has 
been torn out cannot be electrically neutral. If it was 
previously neutral, then it cannot be so now, since it is 
poorer by one elementary charge. That which remains 
must have a positive charge. Furthermore, since the 
mass of an electron is so much smaller than that of the 
lightest atom, we can safely conclude that by far the 
greater part of the mass of the atom is not represented 
by electrons but by the remainder of the elementary 
particles which are much heavier than the electrons. 
We call this heavy part of the atom its nucleus. 

Modern experimental physics has developed methods 


of breaking up the nucleus of the atom, of changing 
atoms of one element into those of another, and of 
extracting from the nucleus the heavy elementary par- 
ticles of which it is built. This chapter of physics, known 
as "nuclear physics", to which Rutherford contributed 
so much, is, from the experimental point of view, the 
most interesting. But a theory, simple in its funda- 
mental ideas and connecting the rich variety of facts in 
the domain of nuclear physics, is still lacking. Since, in 
these pages, we are interested only in general physical 
ideas, we shall omit this chapter in spite of its great 
importance in modern physics. 


Let us consider a wall built along the seashore. The 
waves from the sea continually impinge on the wall, 
wash away some of its surface, and retreat, leaving the 
way clear for the incoming waves. The mass of the wall 
decreases and we can ask how much is washed away 
in, say, one year. But now let us picture a different 
process. We want to diminish the mass of the wall by 
the same amount as previously but in a different way. 
We shoot at the wall and split it at the places where the 
bullets hit. The mass of the wall will be decreased and 
we can well imagine that the same reduction in mass 
is achieved in both cases. But from the appearance of 
the wall we could easily detect whether the continuous 
sea wave or the discontinuous shower of bullets has been 
acting. It will be helpful, in understanding the pheno- 
mena which we are about to describe, to bear in 


mind the difference between sea waves and a shower of 

We said, previously, that a heated wire emits elec- 
trons. Here we shall introduce another way of extracting 
electrons from metal. Homogeneous light, such as 
violet light, which is, as we know, light of a definite 
wave-length, is impinging on a metal surface. The 
light extracts electrons from the metal. The electrons 
are torn from the metal and a shower of them speeds 
along with a certain velocity. From the point of view 
of the energy principle we can say : the energy of light 
is partially transformed into the kinetic energy of 
expelled electrons. Modern experimental technique 
enables us to register these electron-bullets, to determine 
their velocity and thus their energy. This extraction 
of electrons by light falling upon metal is called the 
photoelectric effect. 

Our starting-point was the action of a homogeneous 
light wave, with some definite intensity. As in every 
experiment, we must now change our arrangements 
to see whether this will have any influence on the 
observed effect. 

Let us begin by changing the intensity of the homo- 
geneous violet light falling on the metal plate and 
note to what extent the energy of the emitted electrons 
depends upon the intensity of the light. Let us try to 
find the answer by reasoning instead of by experiment. 
We could argue: in the photoelectric effect a certain 
definite portion of the energy of radiation is transformed 
into energy of motion of the electrons. If we again 


illuminate the metal with light of the same wave-length 
but from a more powerful source, then the energy of 
the emitted electrons should be greater, since the radia- 
tion is richer in energy. We should, therefore, expect 
the velocity of the emitted electrons to increase if the 
intensity of the light increases. But experiment again 
contradicts our prediction. Once more we see that the 
laws of nature are not as we should like them to be. 
We have come upon one of the experiments which, 
contradicting our predictions, breaks the theory on 
which they were based. The actual experimental result 
is, from the point of view of the wave theory, astonishing. 
The observed electrons all have the same speed, the 
same energy, which does not change when the intensity 
of the light is increased. 

This experimental result could not be predicted by 
the wave theory. Here again a new theory arises from 
the conflict between the old theory and experiment. 

Let us be deliberately unjust to the wave theory of 
light, forgetting its great achievements, its splendid 
explanation of the bending of light around very small 
obstacles. With our attention focused on the photo- 
electric effect, let us demand from the theory an 
adequate explanation of this effect. Obviously, we 
cannot deduce from the wave theory the independence 
of the energy of electrons from the intensity of light by 
which they have been extracted from the metal plate. 
We shall, therefore, try another theory. We remember 
that Newton's corpuscular theory, explaining many 
of the observed phenomena of light, failed to account 


for the bending of light, which we are now deliberately 
disregarding. In Newton's time the concept of energy 
did not exist. Light corpuscles were, according to him, 
weightless; each colour preserved its own substance 
character. Later, when the concept of energy was 
created and it was recognized that light carries energy, 
no one thought of applying these concepts to the cor- 
puscular theory of light. Newton's theory was dead 
and, until our own century, its revival was not taken 

To keep the principal idea of Newton's theory, we 
must assume that homogeneous light is composed of 
energy-grains and replace the old light corpuscles by 
light quanta, which we shall call photons, small portions 
of energy, travelling through empty space with the 
velocity of light. The revival of Newton's theory in this 
new form leads to the quantum theory of light. Not only 
matter and electric charge, but also energy of radiation 
has a granular structure, i.e., is built up of light quanta. 
In addition to quanta of matter and quanta of electricity 
there are also quanta of energy. 

The idea of energy quanta was first introduced by 
Planck at the beginning of this century in order to 
explain some effects much more complicated than the 
photoelectric effect. But the photo-effect shows most 
clearly and simply the necessity for changing our old 

It is at once evident that this quantum theory of light 
explains the photoelectric effect. A shower of photons 
is falling on a metal plate. The action between radiation 


and matter consists here of very many single processes 
in which a photon impinges on the atom and tears out 
an electron. These single processes are all alike and the 
extracted electron will have the same energy in every 
case. We also understand that increasing the intensity 
of the light means, in our new language, increasing the 
number of falling photons. In this case, a different 
number of electrons would be thrown out of the metal 
plate, but the energy of any single one would not change. 
Thus we see that this theory is in perfect agreement 
with observation. 

What will happen if a beam of homogeneous light 
of a different colour, say, red instead of violet, falls on 
the metal surface? Let us leave experiment to answer 
this question. The energy of the extracted electrons 
must be measured and compared with the energy of 
electrons thrown out by violet light. The energy of 
the electron extracted by red light turns out to be 
smaller than the energy of the electron extracted by 
violet light. This means that the energy of the light 
quanta is different for different colours. The photons 
belonging to the colour red have half the energy of 
those belonging to the colour violet. Or, more rigor- 
ously: the energy of a light quantum belonging to a 
homogeneous colour decreases proportionally as the 
wave-length increases. There is an essential difference 
between quanta of energy and quanta of electricity. 
Light quanta differ for every wave-length, whereas 
quanta of electricity are always the same. If we were 
to use one of our previous analogies, we should compare 


light quanta to the smallest monetary quanta, differing 
in each country. 

Let us continue to discard the wave theory of light 
and assume that the structure of light is granular and 
is formed by light quanta, that is, photons speeding 
through space with the velocity of light. Thus, in our 
new picture, light is a shower of photons, and the 
photon is the elementary quantum of light energy. If, 
however, the wave theory is discarded, the concept of 
a wave-length disappears. What new concept takes its 
place? The energy of the light quanta! Statements 
expressed in the terminology of the wave theory can be 
translated into statements of the quantum theory of 
radiation. For example: 


Homogeneous light has a Homogeneous light con- 
definite wave-length. The tains photons of a definite 
wave-length of the red end energy. The energy of the 
of the spectrum is twice photon for the red end of 
that of the violet end. the spectrum is half that of 

the violet end. 

The state of affairs can be summarized in the follow- 
ing way : there are phenomena which can be explained 
by the quantum theory but not by the wave theory. 
Photo-effect furnishes an example, though other pheno- 
mena of this kind are known. There are phenomena 
which can be explained by the wave theory but not by 
the quantum theory. The bending of light around 
obstacles is a typical example. Finally, there are 


phenomena, such as the rectilinear propagation of light, 
which can be equally well explained by the quantum 
and the wave theory of light. 

But what is light really? Is it a wave or a shower of 
photons? Once before we put a similar question when 
we asked: is light a wave or a shower of light cor- 
puscles? At that time there was every reason for 
discarding the corpuscular theory of light and accepting 
the wave theory, which covered all phenomena. Now, 
however, the problem is much more complicated. 
There seems no likelihood of forming a consistent de- 
scription of the phenomena of light by a choice of only 
one of the two possible languages. It seems as though 
we must use sometimes the one theory and sometimes 
the other, while at times we may use either. We are 
faced with a new kind of difficulty. We have two 
contradictory pictures of reality; separately neither 
of them fully explains the phenomena of light, but 
together they do ! 

How is it possible to combine these two pictures? 
How can we understand these two utterly different 
aspects of light? It is not easy to account for this new 
difficulty. Again we are faced with a fundamental 

For the moment let us accept the photon theory of 
light and try, by its help, to understand the facts so 
far explained by the wave theory. In this way we shall 
stress the difficulties which make the two theories 
appear, at first sight, irreconcilable. 

We remember : a beam of homogeneous light passing 


through a pinhole gives light and dark rings (p. 118). 
How is it possible to understand this phenomenon by 
the help of the quantum theory of light, disregarding 
the wave theory? A photon passes through the hole. 
We could expect the screen to appear light if the photon 
passes through and dark if it does not. Instead, we 
find light and dark rings. We could try to account for 
it as follows : perhaps there is some interaction between 
the rim of the hole and the photon which is responsible 
for the appearance of the diffraction rings. This sentence 
can, of course, hardly be regarded as an explanation. 
At best, it outlines a programme for an explanation 
holding out at least some hope of a future under- 
standing of diffraction by interaction between matter 
and photons. 

But even this feeble hope is dashed by our previous 
discussion of another experimental arrangement. Let us 
take two pinholes. Homogeneous light passing through 
the two holes gives light and dark stripes on the screen. 
How is this effect to be understood from the point of 
view of the quantum theory of light? We could argue: 
a photon passes through either one of the two pinholes. 
If a photon of homogeneous light represents an elemen- 
tary light particle, we can hardly imagine its division and 
its passage through the two holes. But then the effect 
should be exactly as in the first case, light and dark 
rings and not light and dark stripes. How is it possible 
then that the presence of another pinhole completely 
changes the effect? Apparently the hole through 
which the photon does not pass, even though it may be 


at a fair distance, changes the rings into stripes ! If 
the photon behaves like a corpuscle in classical physics, 
it must pass through one of the two holes. But in this 
case, the phenomena of diffraction seem quite incom- 

Science forces us to create new ideas, new theories. 
Their aim is to break down the wall of contradictions 
which frequently blocks the way of scientific progress. 
All the essential ideas in science were born in a dramatic 
conflict between reality and our attempts at under- 
standing. Here again is a problem for the solution of 
which new principles are needed. Before we try to 
account for the attempts of modern physics to explain 
the contradiction between the quantum and the wave 
aspects of light, we shall show that exactly the same 
difficulty appears when dealing with quanta of matter 
instead of quanta of light. 


We already know that all matter is built of only a 
few kinds of particles. Electrons were the first elemen- 
tary particles of matter to be discovered. But electrons 
are also the elementary quanta of negative electricity. 
We learned furthermore that some phenomena force 
us to assume that light is composed of elementary light 
quanta, differing for different wave-lengths. Before 
proceeding we must discuss some physical phenomena in 
which matter as well as radiation plays an essential role. 
The sun emits radiation which can be split into its 
components by a prism. The continuous spectrum of 


the sun can thus be obtained. Every wave-length be- 
tween the two ends of the visible spectrum is repre- 
sented. Let us take another example. It was previously 
mentioned that sodium when incandescent emits homo- 
geneous light, light of one colour or one wave-length. 
If incandescent sodium is placed before the prism, we 
see only one yellow line. In general, if a radiating body 
is placed before the prism, then the light it emits is 
split up into its components, revealing the spectrum 
characteristic of the emitting body. 

The discharge of electricity in a tube containing gas 
produces a source of light such as seen in the neon 
tubes used for luminous advertisements. Suppose such 
a tube is placed before a spectroscope. The spectroscope 
is an instrument which acts like a prism, but with much 
greater accuracy and sensitiveness; it splits light into 
its components, that is, it analyses it. Light from the 
sun, seen through a spectroscope, gives a continuous 
spectrum; all wave-lengths are represented in it. If, 
however, the source of light is a gas through which a 
current of electricity passes, the spectrum is of a different 
character. Instead of the continuous, multi-coloured 
design of the sun's spectrum, bright, separated stripes 
appear on a continuous dark background. Every stripe, 
if it is very narrow, corresponds to a definite colour or, 
in the language of the wave theory, to a definite wave- 
length. For example, if twenty lines are visible in the 
spectrum, each of them will be designated by one of 
twenty numbers expressing the corresponding wave- 
length. The vapours of the various elements possess 


different systems of lines, and thus different combinations 
of numbers designating the wave-lengths composing the 
emitted light spectrum. No two elements have identical 
systems of stripes in their characteristic spectra, just 
as no two persons have exactly identical finger-prints. As 
a catalogue of these lines was worked out by physicists, 
the existence of laws gradually became evident, and 
it was possible to replace some of the columns of 
seemingly disconnected numbers expressing the length of 
the various waves by one simple mathematical formula. 

All that has just been said can now be translated into 
the photon language. The stripes correspond to certain 
definite wave-lengths or, in other words, to photons with 
a definite energy. Luminous gases do not, therefore, 
emit photons with all possible energies, but only those 
characteristic of the substance. Reality again limits the 
wealth of possibilities. 

Atoms of a particular element, say, hydrogen, can 
emit only photons with definite energies. Only the 
emission of definite energy quanta is permissible, all 
others being prohibited. Imagine, for the sake of sim- 
plicity, that some element emits only one line, that is, 
photons of a quite definite energy. The atom is richer 
in energy before the emission and poorer afterwards. 
From the energy principle it must follow that the energy 
level of an atom is higher before emission and lower 
afterwards, and that the difference between the two 
levels must be equal to the energy of the emitted photon. 
Thus the fact that an atom of a certain element emits 
radiation of one wave-length only, that is photons of a 


definite energy only, could be expressed differently: 
only two energy levels are permissible in an atom of 
this element and the emission of a photon corresponds 
to the transition of the atom from the higher to the 
lower energy level. 

But more than one line appears in the spectra of the 
elements, as a rule. The photons emitted correspond to 
many energies and not to one only. Or, in other words, 
we must assume that many energy levels are allowed in 
an atom and that the emission of a photon corresponds 
to the transition of the atom from a higher energy level 
to a lower one. But it is essential that not every energy 
level should be permitted, since not every wave-length, 
not every photon-energy, appears in the spectra of an 
element. Instead of saying that some definite lines, 
some definite wave-lengths, belong to the spectrum of 
every atom, we can say that every atom has some definite 
energy levels, and that the emission of light quanta 
is associated with the transition of the atom from one 
energy level to another. The energy levels are, as a rule, 
not continuous but discontinuous. Again we see that 
the possibilities are restricted by reality. 

It was Bohr who showed for the first time why just 
these and no other lines appear in the spectra. His 
theory, formulated twenty-five years ago, draws a pic- 
ture of the atom from which, at any rate in simple cases, 
the spectra of the elements can be calculated and the 
apparently dull and unrelated numbers are suddenly 
made coherent in the light of the theory. 

Bohr's theory forms an intermediate step toward a 


deeper and more general theory, called the wave or 
quantum mechanics. It is our aim in these last pages to 
characterize the principal ideas of this theory. Before 
doing so, we must mention one more theoretical and 
experimental result of a more special character. 

Our visible spectrum begins with a certain wave- 
length for the violet colour and ends with a certain 
wave-length for the red colour. Or, in other words, 
the energies of the photons in the visible spectrum are 
always enclosed within the limits formed by the photon 
energies of the violet and red lights. This limitation is, 
of course, only a property of the human eye. If the 
difference in energy of some of the energy levels is 
sufficiently great, then an ultraviolet photon will be sent 
out, giving a line beyond the visible spectrum. Its 
presence cannot be detected by the naked eye; a photo- 
graphic plate must be used. 

X-rays are also composed of photons of a much 
greater energy than those of visible light, or in other 
words, their wave-lengths are much smaller, thousands 
of times smaller in fact, than those of visible light. 

But is it possible to determine such small wave- 
lengths experimentally? It was difficult enough to do 
so for ordinary light. We had to have small obstacles or 
small apertures. Two pinholes very near to each other, 
showing diffraction for ordinary light, would have to be 
many thousands of times smaller and closer together to 
show diffraction for X-rays. 

How then can we measure the wave-lengths of these 
rays? Nature herself comes to our aid. 


A crystal is a conglomeration of atoms arranged at 
very short distances from each other on a perfectly 
regular plan. Our drawing shows a simple model of 
the structure of a crystal. Instead of minute apertures, 
there are extremely small obstacles formed by the atoms 

of the element, arranged very close to each other in 
absolutely regular order. The distances between the 
atoms, as found from the theory of the crystal structure, 
are so small that they might be expected to show the 
effect of diffraction for X-rays. Experiment proved that 
it is, in fact, possible to diffract the X-ray wave by 
means of these closely packed obstacles disposed in the 
regular three-dimensional arrangement occurring in a 

Suppose that a beam of X-rays falls upon a crystal 
and, after passing through it, is recorded on a photo- 


graphic plate. The plate then shows the diffraction 
pattern. Various methods have been used to study the 
X-ray spectra, to deduce data concerning the wave- 
length from the diffraction pattern. What has been said 
here in a few words would fill volumes if all theoretical 
and experimental details were set forth. In Plate III 
we give only one diffraction pattern obtained by one 
of the various methods. We again see the dark and light 
rings so characteristic of the wave theory. In the centre 
the non-diffracted ray is visible. If the crystal were not 
brought between the X-rays and the photographic 
plate, only the light spot in the centre would be seen. 
From photographs of this kind the wave-lengths of the 
X-ray spectra can be calculated and, on the other hand, 
if the wave-length is known, conclusions can be drawn 
about the structure of the crystal. 


How can we understand the fact that only certain 
characteristic wave-lengths appear in the spectra of the 

It has often happened in physics that an essential 
advance was achieved by carrying out a consistent 
analogy between apparently unrelated phenomena. In 
these pages we have often seen how ideas created and 
developed in one branch of science were afterwards 
successfully applied to another. The development of 
the mechanical and field views gives many examples 
of this kind. The association of solved problems with 
those unsolved may throw new light on our difficulties 


I: i 



/ *- 





A 1 



O 1 * 


: ill 



(Photographed by A. G". Shenslone) 
Spectral lines 

(Photographed by iMstowiecki and Gregor] 

Diffraction of X-rays 

(Photographed by Ijoria and Klinger) 
Diffraction of electronic waves 


by suggesting new ideas. It is easy to find a superficial 
analogy which really expresses nothing. But to dis- 
cover some essential common features, hidden beneath 
a surface of external differences, to form, on this basis, 
a new successful theory, is important creative work. 
The development of the so-called wave mechanics, 
begun by de Broglie and Schrodinger, less than fifteen 
years ago, is a typical example of the achievement of a 
successful theory by means of a deep and fortunate 

Our starting-point is a classical example having 
nothing to do with modern physics. We take in our 

hand the end of a very long flexible rubber tube, or a 
very long spring, and try to move it rhythmically up 
and down, so that the end oscillates. Then, as we have 
seen in many other examples, a wave is created by the 
oscillation which spreads through the tube with a certain 
velocity. If we imagine an infinitely long tube, then the 
portions of waves, once started, will pursue their endless 
journey without interference. 

Now another case. The two ends of the same tube 
are fastened. If preferred, a violin string may be used. 
What happens now if a wave is created at one end of 
the rubber tube or cord? The wave begins its journey 
as in the previous example, but it is soon reflected by 
the other end of the tube. We now have two waves : 


one creation by oscillation, the other by reflection; they 
travel in opposite directions and interfere with each 
other. It would not be difficult to trace the interference 
of the two waves and discover the one wave resulting 
from their superposition; it is called the standing wave. 
The two words "standing" and "wave" seem to con- 
tradict each other; their combination is, nevertheless, 
justified by the result of the superposition of the two 

The simplest example of a standing wave is the 
motion of a cord with the two ends fixed, an up-and- 
down motion, as shown in our drawing. This motion is 

the result of one wave lying on the other when the two 
are travelling in opposite directions. The characteristic 
feature of this motion is : only the two end-points are 
at rest. They are called nodes. The wave stands, so to 
speak, between the two nodes, all points of the cord 
reaching simultaneously the maxima and minima of 
their deviation. 

But this is only the simplest kind of a standing wave. 
There are others. For example, a standing wave can 
have three nodes, one at each end and one in the centre. 


In this case three points are always at rest. A glance at 
the drawings shows that here the wave-length is half 
as great as the one with two nodes. Similarly, standing 

waves can have four, five, and more nodes. The wave- 
length in each case will depend on the number of 
nodes. This number can only be an integer and can 
change only by jumps. The sentence "the number of 

nodes in a standing wave is 3*576" is pure nonsense. 
Thus the wave-length can only change discontinuously. 
Here, in this most classical problem, we recognize the 
familiar features of the quantum theory. The standing 
wave produced by a violin player is, in fact, still more 
complicated, being a mixture of very many waves with 
two, three, four, five, and more nodes and, therefore, 
a mixture of several wave-lengths. Physics can analyse 
such a mixture into the simple standing waves from 
which it is composed. Or, using our previous ter- 
minology, we could say that the oscillating string has 
its spectrum, just as an element emitting radiation. 
And, in the same way as for the spectrum of an element, 
only certain wave-lengths are allowed, all others being 


We have thus discovered some similarity between 
the oscillating cord and the atom emitting radiation. 
Strange as this analogy may seem, let us draw further 
conclusions from it and try to proceed with the com- 
parison, once having chosen it. The atoms of every 
element are composed of elementary particles, the 
heavier constituting the nucleus, and the lighter the 
electrons. Such a system of particles behaves like a 
small acoustical instrument in which standing waves 
are produced. 

Yet the standing wave is the result of interference 
between two or, generally, even more moving waves. 
If there is some truth in our analogy, a still simpler 
arrangement than that of the atom should correspond 
to a spreading wave. What is the simplest arrangement? 
In our material world, nothing can be simpler than an 
electron, an elementary particle, on which no forces are 
acting, that is, an electron at rest or in uniform motion. 
We could guess a further link in the chain of our 
analogy: electron moving uniformly -> waves of a 
definite length. This was de Broglie's new and coura- 
geous idea. 

It was previously shown that there are phenomena in 
which light reveals its wave-like character and others 
in which light reveals its corpuscular character. After 
becoming used to the idea that light is a wave, we 
found, to our . astonishment, that in some cases, for 
instance in the photoelectric effect, it behaves like a 
shower of photons. Now we have just the opposite 
state of affairs for electrons. We accustomed ourselves 


to the idea that electrons are particles, elementary 
quanta of electricity and matter. Their charge and mass 
were investigated. If there is any truth in de Broglie's 
idea, then there must be some phenomena in which 
matter reveals its wave-like character. At first, this 
conclusion, reached by following the acoustical analogy, 
seems strange and incomprehensible. How can a moving 
corpuscle have anything to do with a wave? But this is 
not the first time we have faced a difficulty of this kind 
in physics. We met the same problem in the domain of 
light phenomena. 

Fundamental ideas play the most essential role in 
forming a physical theory. Books on physics are full 
of complicated mathematical formulae. But thought 
and ideas, not formulae, are the beginning of every 
physical theory. The ideas must later take the mathe- 
matical form of a quantitative theory, to make possible 
the comparison with experiment. This can be explained 
by the example of the problem with which we are now 
dealing. The principal guess is that the uniformly 
moving electron will behave, in some phenomena, like a 
wave. Assume that an electron or a shower of electrons, 
provided they all have the same velocity, is moving 
uniformly. The mass, charge, and velocity of each in- 
dividual electron are known. If we wish to associate in 
some way a wave concept with a uniformly moving 
electron or electrons, our next question must be: what 
is the wave-length? This is a quantitative question and 
a more or less quantitative theory must be built up to 
answer it. This is indeed a simple matter. The mathe- 



matical simplicity of de Broglie's work, providing an 
answer to this question, is most astonishing. At the 
time his work was done, the mathematical technique 
of other physical theories was very subtle and compli- 
cated, comparatively speaking. The mathematics deal- 
ing with the problem of waves of matter is extremely 
simple and elementary but the fundamental ideas are 
deep and far-reaching. 

Previously, in the case of light waves and photons, 
it was shown that every statement formulated in the 
wave language can be translated into the language 
of photons or light corpuscles. The same is true for 
electronic waves. For uniformly moving electrons, the 
corpuscular language is already known. But every 
statement expressed in the corpuscular language can 
be translated into the wave language, just as in the case 
of photons. Two clues laid down the rules of transla- 
tion. The analogy between light waves and electronic 
waves or photons and electrons is one clue. We try to 
use the same method of translation for matter as for 
light. The special relativity theory furnished the other 
clue. The laws of nature must be invariant with respect 
to the Lorentz and not to the classical transformation. 
These two clues together determine the wave-length 
corresponding to a moving electron. It follows from 
the theory that an electron moving with a velocity of, 
say, 10,000 miles per second, has a wave-length which 
can be easily calculated, and which turns out to lie in 
the same region as the X-ray wave-lengths. Thus we 
conclude further that if the wave character of matter 


can be detected, it should be done experimentally in an 
analogous way to that of X-rays. 

Imagine an electron beam moving uniformly with 
a given velocity, or, to use the wave terminology, a 
homogeneous electronic wave, and assume that it falls 
on a very thin crystal, playing the part of a diffraction 
grating. The distances between the diffracting ob- 
stacles in the crystal are so small that diffraction for 
X-rays can be produced. One might expect a similar 
effect for electronic waves with the same order of 
wave-length. A photographic plate would register this 
diffraction of electronic waves passing through the thin 
layer of crystal. Indeed, the experiment produces what 
is undoubtedly one of the great achievements of the 
theory: the phenomenon of diffraction for electronic 
waves. The similarity between the diffraction of an 
electronic wave and that of an X-ray is particularly 
marked as seen from a comparison of the patterns in 
Plate III. We know that such pictures enable us to 
determine the wave-lengths of X-rays. The same holds 
good for electronic waves. The diffraction pattern 
gives the length of a wave of matter and the perfect 
quantitative agreement between theory and experiment 
confirms the chain of our argument splendidly. 

Our previous difficulties are broadened and deep- 
ened by this result. This can be made clear by an ex- 
ample similar to the one given for a light wave. An 
electron shot at a very small hole will bend like a light 
wave. Light and dark rings appear on the photo- 
graphic plate. There may be some hope of explaining 


this phenomenon by the interaction between the elec- 
tron and the rim, though such an explanation does not 
seem to be very promising. But what about the two 
pinholes? Stripes appear instead of rings. How is it 
possible that the presence of the other hole completely 
changes the effect? The electron is indivisible and can, 
it would seem, pass through only one of the two holes. 
How could an electron passing through a hole possibly 
know that another hole has been made some distance 

We asked before : what is light? Is it a shower of cor- 
puscles or a wave? We now ask: what is matter, what 
is an electron? Is it a particle or a wave? The electron 
behaves like a particle when moving in an external 
electric or magnetic field. It behaves like a wave when 
diffracted by a crystal. With the elementary quanta of 
matter we came across the same difficulty that we met 
with in the light quanta. One of the most fundamental 
questions raised by recent advance in science is how 
to reconcile the two contradictory views of matter and 
wave. It is one of those fundamental difficulties which, 
once formulated, must lead, in the long run, to scien- 
tific progress. Physics has tried to solve this problem. 
The future must decide whether the solution sug- 
gested by modern physics is enduring or temporary. 


If, according to classical mechanics, we know the 
position and velocity of a given material point and also 
what external forces are acting, we can predict, from 


the mechanical laws, the whole of its future path. The 
sentence: "The material point has such-and-such posi- 
tion and velocity at such-and-such an instant," has a 
definite meaning in classical mechanics. If this state- 
ment were to lose its sense, our argument (p. 32) about 
foretelling the future path would fail. 

In the early nineteenth century, scientists wanted to 
reduce all physics to simple forces acting on material 
particles that have definite positions and velocities at 
any instant. Let us recall how we described motion 
when discussing mechanics at the beginning of our 
journey through the realm of physical problems. We 
drew points along a definite path showing the exact 
positions of the body at certain instants and then 
tangent vectors showing the direction and magnitude 
of the velocities. This was both simple and convincing. 
But it cannot be repeated for our elementary quanta of 
matter, that is electrons, or for quanta of energy, that 
is photons. We cannot picture the journey of a photon 
or electron in the way we imagined motion in classical 
mechanics. The example of the two pinholes shows this 
clearly. Electron and photon seem to pass through the 
two holes. It is thus impossible to explain the effect by 
picturing the path of an electron or a photon in the old 
classical way. 

We must, of course, assume the presence of ele- 
mentary actions, such as the passing of electrons or 
photons through the holes. The existence of elementary 
quanta of matter and energy cannot be doubted. But 
the elementary laws certainly cannot be formulated by 


specifying positions and velocities at any instant in the 
simple manner of classical mechanics. 

Let us, therefore, try something different. Let us 
continually repeat the same elementary processes. One 
after the other, the electrons are sent in the direction of 
the pinholes. The word "electron" is used here for the 
sake of definiteness; our argument is also valid for 

The same experiment is repeated over and over again 
in exactly the same way; the electrons all have the same 
velocity and move in the direction of the two pinholes. 
It need hardly be mentioned that this is an idealized 
experiment which cannot be carried out in reality but 
may well be imagined. We cannot shoot out single 
photons or electrons at given instants, like bullets from 
a gun. 

The outcome of repeated experiments must again be 
dark and light rings for one hole and dark and light 
stripes for two. But there is one essential difference. In 
the case of one individual electron, the experimental 
result was incomprehensible. It is more easily under- 
stood when the experiment is repeated many times. We 
can now say: light stripes appear where many electrons 
fall. The stripes become darker at the place where 
fewer electrons are falling. A completely dark spot 
means that there are no electrons. We are not, of 
course, allowed to assume that all the electrons pass 
through one of the holes. If this were so, it could not 
make the slightest difference whether or not the other 
is covered. But we already know that covering the 


second hole does make a difference. Since one particle 
is indivisible, we cannot imagine that it passes through 
both the holes. The fact that the experiment was re- 
peated many times points to another way out. Some of 
the electrons may pass through the first hole and others 
through the second. We do not know why individual 
electrons choose particular holes, but the net result of 
repeated experiments must be that both pinholes par- 
ticipate in transmitting the electrons from the source 
to the screen. If we state only what happens to the 
crowd of electrons when the experiment is repeated, not 
bothering about the behaviour of individual particles, 
the difference between the ringed and the striped 
pictures becomes comprehensible. By the discussion 
of a sequence of experiments a new idea was born, 
that of a crowd with the individuals behaving in an 
unpredictable way. We cannot foretell the course of 
one single electron, but we can predict that, in the net 
result, the light and dark stripes will appear on the 

Let us leave quantum physics for the moment. 

We have seen in classical physics that if we know 
the position and velocity of a material point at a certain 
instant and the forces acting upon it, we can predict its 
future path. We also saw how the mechanical point of 
view was applied to the kinetic theory of matter. But 
in this theory a new idea arose from our reasoning. It 
will be helpful in understanding later arguments to 
grasp this idea thoroughly. 

There is a vessel containing gas. In attempting to 


trace the motion of every particle one would have to 
commence by finding the initial states, that is, the 
initial positions and velocities of all the particles. Even 
if this were possible, it would take more than a human 
lifetime to set down the result on paper, owing to the 
enormous number of particles which would have to be 
considered. If one then tried to employ the known 
methods of classical mechanics for calculating the final 
positions of the particles, the difficulties would be 
insurmountable. In principle, it is possible to use the 
method applied for the motion of planets, but in prac- 
tice this is useless and must give way to the method of 
statistics. This method dispenses with any exact know- 
ledge of initial states. We know less about the system at 
any given moment and are thus less able to say any- 
thing about its past or future. We become indifferent 
to the fate of the individual gas particles. Our problem 
is of a different nature. For example : we do not ask, 
"What is the speed of every particle at this moment?" 
But we may ask: "How many particles have a speed 
between 1000 and uoo feet per second?" We care 
nothing for individuals. What we seek to determine 
are average values typifying the whole aggregation. It 
is clear that there can be some point in a statistical 
method of reasoning only when the system consists of 
a large number of individuals. 

By applying the statistical method we cannot fore- 
tell the behaviour of an individual in a crowd. We can 
only foretell the chance, the probability, that it will 
behave in some particular manner. If our statistical 


laws tell us that one-third of the particles have a speed 
between 1000 and uoo feet per second, it means that 
by repeating our observations for many particles, we 
shall really obtain this average, or in other words, that 
the probability of finding a particle within this limit is 
equal to one-third. 

Similarly, to know the birth rate of a great com- 
munity does not mean knowing whether any particular 
family is blessed with a child. It means a knowledge of 
statistical results in which the contributing personalities 
play no role. 

By observing the registration plates of a great many 
cars we can soon discover that one-third of their num- 
bers are divisible by three. But we cannot foretell 
whether the car which will pass in the next moment 
will have this property. Statistical laws can be applied 
only to big aggregations, but not to their individual 

We can now return to our quantum problem. 

The laws of quantum physics are of a statistical 
character. This means: they concern not one single 
system but an aggregation of identical systems; they 
cannot be verified by measurement of one individual, 
but only by a series of repeated measurements. 

Radioactive disintegration is one of the many events 
for which quantum physics tries to formulate laws 
governing the spontaneous transmutation from one ele- 
ment to another. We know, for example, that in 1600 
years half of one gram of radium will disintegrate, and 
half will remain. We can foretell approximately how 


many atoms will disintegrate during the next half-hour, 
but we cannot say, even in our theoretical descrip- 
tions, why just these particular atoms are doomed. 
According to our present knowledge, we have no 
power to designate the individual atoms condemned to 
disintegration. The fate of an atom does not depend on 
its age. There is not the slightest trace of a law governing 
their individual behaviour. Only statistical laws can 
be formulated, laws governing large aggregations of 

Take another example. The luminous gas of some 
element placed before a spectroscope shows lines of 
definite wave-length. The appearance of a discontinu- 
ous set of definite wave-lengths is characteristic of the 
atomic phenomena in which the existence of element- 
ary quanta is revealed. But there is still another aspect 
of this problem. Some of the spectrum lines are very 
distinct, others are fainter. A distinct line means that a 
comparatively large number of photons belonging to 
this particular wave-length are emitted; a faint line 
means that a comparatively small number of photons 
belonging to this wave-length are emitted. Theory 
again gives us statements of a statistical nature only. 
Every line corresponds to a transition from higher to 
lower energy level. Theory tells us only about the 
probability of each of these possible transitions, but 
nothing about the actual transition of an individual 
atom. The theory works splendidly because all these 
phenomena involve large aggregations and not single 


It seems that the new quantum physics resembles 
somewhat the kinetic theory of matter, since both are 
of a statistical nature and both refer to great aggrega- 
tions. But this is not so ! In this analogy an understanding 
not only of the similarities but also of the differences 
is most important. The similarity between the kinetic 
theory of matter and quantum physics lies chiefly in 
their statistical character. But what are the differences? 

If we wish to know how many men and women over 
the age of twenty live in a city, we must get every 
citizen to fill up a form under the headings "male", 
"female " , and ' c age ' ' . Provided every answer is correct, 
we can obtain, by counting and segregating them, a 
result of a statistical nature. The individual names and 
addresses on the forms are of no account. Our statis- 
tical view is gained by the knowledge of individual 
cases. Similarly, in the kinetic theory of matter, we 
have statistical laws governing the aggregation, gained 
on the basis of individual laws. 

But in quantum physics the state of affairs is entirely 
different. Here the statistical laws are given immedi- 
ately. The individual laws are discarded. In the ex- 
ample of a photon or an electron and two pinholes we 
have seen that we cannot describe the possible motion 
of elementary particles in space and time as we did in 
classical physics. Quantum physics abandons individual 
laws of elementary particles and states directly the sta- 
tistical laws governing aggregations. It is impossible, on 
the basis of quantum physics, to describe positions and 
velocities of an elementary particle or to predict its 


future path as in classical physics. Quantum physics 
deals only with aggregations, and its laws are for 
crowds and not for individuals. 

It is hard necessity and not speculation or a desire for 
novelty which forces us to change the old classical 
view. The difficulties of applying the old view have 
been outlined for one instance only, that of diffraction 
phenomena. But many others, equally convincing, 
could be quoted. Changes of view are continually 
forced upon us by our attempts to understand reality. 
But it always remains for the future to decide whether 
we chose the only possible way out and whether or not a 
better solution of our difficulties could have been found. 

We have had to forsake the description of individual 
cases as objective happenings in space and time; we 
have had to introduce laws of a statistical nature. These 
are the chief characteristics of modern quantum 

Previously, when introducing new physical realities, 
such as the electromagnetic and gravitational field, we 
tried to indicate in general terms the characteristic 
features of the equations through which the ideas 
have been mathematically formulated. We shall now 
do the same with quantum physics, referring only very 
briefly to the work of Bohr, de Broglie, Schrodinger, 
Heisenberg, Dirac and Born. 

Let us consider the case of one electron. The elec- 
tron may be under the influence of an arbitrary foreign 
electromagnetic field, or free from all external influences. 
It may move, for instance, in the field of an atomic 


nucleus or it may diffract on a crystal. Quantum 
physics teaches us how to formulate the mathematical 
equations for any of these problems. 

We have already recognized the similarity between 
an oscillating cord, the membrane of a drum, a wind 
instrument, or any other acoustical instrument on the 
one hand, and a radiating atom on the other. There is 
also some similarity between the mathematical equa- 
tions governing the acoustical problem and those gov- 
erning the problem of quantum physics. But again the 
physical interpretation of the quantities determined in 
these two cases is quite different. The physical quan- 
tities describing the oscillating cord and the radiating 
atom have quite a different meaning, despite some 
formal likeness in the equations. In the case of the cord, 
we ask about the deviation of an arbitrary point from 
its normal position at an arbitrary moment. Knowing 
the form of the oscillating cord at a given instant, we 
know everything we wish. The deviation from the 
normal can thus be calculated for any other moment 
from the mathematical equations for the oscillating 
cord. The fact that some definite deviation from the 
normal position corresponds to every point of the cord 
is expressed more rigorously as follows: for any in- 
stant, the deviation from the normal value is a function 
of the co-ordinates of the cord. All points of the cord 
form a one-dimensional continuum, and the deviation 
is a function defined in this one-dimensional con- 
tinuum, to be calculated from the equations of the 
oscillating cord. 


Analogously, in the case of an electron a certain 
function is determined for any point in space and for 
any moment. We shall call this function the probability 
wave. In our analogy the probability wave corresponds 
to the deviation from the normal position in the acous- 
tical problem. The probability wave is, at a given in- 
stant, a function of a three-dimensional continuum, 
whereas, in the case of the cord the deviation was, at a 
given moment, a function of the one-dimensional con- 
tinuum. The probability wave forms the catalogue of 
our knowledge of the quantum system under considera- 
tion and will enable us to answer all sensible statistical 
questions concerning this system. It does not tell us 
the position and velocity of the electron at any moment 
because such a question has no sense in quantum 
physics. But it will tell us the probability of meeting 
the electron on a particular spot, or where we have the 
greatest chance of meeting an electron. The result does 
not refer to one, but to many repeated measurements. 
Thus the equations of quantum physics determine the 
probability wave just as Maxwell's equations determine 
the electromagnetic field and the gravitational equa- 
tions determine the gravitational field. The laws of 
quantum physics are again structure laws. But the 
meaning of physical concepts determined by these 
equations of quantum physics is much more abstract 
than in the case of electromagnetic and gravitational 
fields; they provide only the mathematical means of 
answering questions of a statistical nature. 

So far we have considered the electron in some ex- 


ternal field. If it were not the electron, the smallest 
possible charge, but some respectable charge containing 
billions of electrons, we could disregard the whole 
quantum theory and treat the problem according to 
our old pre-quantum physics. Speaking of currents in 
a wire, of charged conductors, of electromagnetic 
waves, we can apply our old simple physics contained 
in Maxwell's equations. But we cannot do this when 
speaking of the photoelectric effect, intensity of spec- 
tral lines, radioactivity, diffraction of electric waves 
and many other phenomena in which the quantum 
character of matter and energy is revealed. We must 
then, so to speak, go one floor higher. Whereas in 
classical physics we spoke of positions and velocities of 
one particle, we must now consider probability waves, 
in a three-dimensional continuum corresponding to 
this one-particle problem. 

Quantum physics gives its own prescription for 
treating a problem if we have previously been taught 
how to treat an analogous problem from the point of 
view of classical physics. 

For one elementary particle, electron or photon, we 
have probability waves in a three-dimensional con- 
tinuum, characterizing the statistical behaviour of the 
system if the experiments are often repeated. But what 
about the case of not one but two interacting particles, 
for instance, two electrons, electron and photon, or 
electron and nucleus? We cannot treat them separately 
and describe each of them through a probability wave 
in three dimensions, just because of their mutual inter- 


action. Indeed, it is not very difficult to guess how to 
describe in quantum physics a system composed of two 
interacting particles. We have to descend one floor, to 
return for a moment to classical physics. The position 
of two material points in space, at any moment, is cha- 
racterized by six numbers, three for each of the points. 
All possible positions of the two material points form a 
six-dimensional continuum and not a three-dimensional 
one as in the case of one point. If we now again ascend 
one floor, to quantum physics, we shall have prob- 
ability waves in a six-dimensional continuum and not in 
a three-dimensional continuum as in the case of one 
particle. Similarly, for three, four, and more particles 
the probability waves will be functions in a continuum 
of nine, twelve, and more dimensions. 

This shows clearly that the probability waves are 
more abstract than the electromagnetic and gravita- 
tional field existing and spreading in our three-dimen- 
sional space. The continuum of many dimensions forms 
the background for the probability waves, and only 
for one particle does the number of dimensions equal 
that of physical space. The only physical significance 
of the probability wave is that it enables us to answer 
sensible statistical questions in the case of many par- 
ticles as well as of one. Thus, for instance, for one 
electron we could ask about the probability of meeting 
an electron in some particular spot. For two particles 
our question could be: what is the probability of meet- 
ing the two particles at two definite spots at a given 


Our first step away from classical physics was aban- 
doning the description of individual cases as objective 
events in space and time. We were forced to apply the 
statistical method provided by the probability waves. 
Once having chosen this way, we are obliged to go 
further toward abstraction. Probability waves in many 
dimensions corresponding to the many-particle pro- 
blem must be introduced. 

Let us, for the sake of briefness, call everything ex- 
cept quantum physics, classical physics. Classical and 
quantum physics differ radically. Classical physics aims 
at a description of objects existing in space, and the 
formulation of laws governing their changes in time. 
But the phenomena revealing the particle and wave 
nature of matter and radiation, the apparently statistical 
character of elementary events such as radioactive dis- 
integration, diffraction, emission of spectral lines, and 
many others, forced us to give up this view. Quantum 
physics does not aim at the description of individual 
objects in space and their changes in time. There is no 
place in quantum physics for statements such as: "This 
object is so-and-so, has this-and-this property.' 5 Instead 
we have statements of this kind: "There is such-and- 
such a probability that the individual object is so-and-so 
and has this-and-this property.' 5 There is no place in 
quantum physics for laws governing the changes in 
time of the individual object. Instead, we have laws 
governing the changes in time of the probability. 
Only this fundamental change, brought into physics 
by the quantum theory, made possible an adequate 


explanation of the apparently discontinuous and statis- 
tical character of events in the realm of phenomena in 
which the elementary quanta of matter and radiation 
reveal their existence. 

Yet new, still more difficult problems arise which 
have not been definitely settled as yet. We shall men- 
tion only some of these unsolved problems. Science is 
not and will never be a closed book. Every important 
advance brings new questions. Every development re- 
veals, in the long run, new and deeper difficulties. 

We already know that in the simple case of one or 
many particles we can rise from the classical to the 
quantum description, from the objective description of 
events in space and time to probability waves. But we 
remember the all-important field concept in classical 
physics. How can we describe interaction between 
elementary quanta of matter and field? If a probability 
wave in thirty dimensions is needed for the quantum 
description of ten particles, then a probability wave 
with an infinite number of dimensions would be needed 
for the quantum description of a field. The transition 
from the classical field concept to the corresponding 
problem of probability waves in quantum physics is a 
very difficult step. Ascending one floor is here no easy 
task and all attempts so far made to solve the problem 
must be regarded as unsatisfactory. There is also one 
other fundamental problem. In all our arguments about 
the transition from classical physics to quantum physics 
we used the old pre-relativistic description in which 
space and time are treated differently. If, however, we 


try to begin from the classical description as proposed 
by the relativity theory, then our ascent to the quan- 
tum problem seems much more complicated. This is 
another problem tackled by modern physics, but still 
far from a complete and satisfactory solution. There is 
still a further difficulty in forming a consistent physics 
for heavy particles, constituting the nuclei. In spite of 
the many experimental data and the many attempts to 
throw light on the nuclear problem, we are still in the 
dark about some of the most fundamental questions in 
this domain. 

There is no doubt that quantum physics explained a 
very rich variety of facts, achieving, for the most part, 
splendid agreement between theory and observation. 
The new quantum physics removes us still further 
from the old mechanical view, and a retreat to the 
former position seems, more than ever, unlikely. But 
there is also no doubt that quantum physics must still 
be based on the two concepts: matter and field. It is, 
in this sense, a dualistic theory and does not bring our 
old problem of reducing everything to the field con- 
cept even one step nearer realization. 

Will the further development be along the line 
chosen in quantum physics, or is it more likely that 
new revolutionary ideas will be introduced into physics? 
Will the road of advance again make a sharp turn, as it 
has so often done in the past? 

During the last few years all the difficulties of quan- 
tum physics have been concentrated around a few 
principal points. Physics awaits their solution impa- 


tiently. But there is no way of foreseeing when and 
where the clarification of these difficulties will be 
brought about. 


What are the general conclusions which can be drawn 
from the development of physics indicated here in a 
broad outline representing only the most fundamental 

Science is not just a collection of laws, a catalogue of 
unrelated facts. It is a creation of the human mind, 
with its freely invented ideas and concepts. Physical 
theories try to form a picture of reality and to estab- 
lish its connection with the wide world of sense im- 
pressions. Thus the only justification for our mental 
structures is whether and in what way our theories 
form such a link. 

We have seen new realities created by the advance 
of physics. But this chain of creation can be traced 
back far beyond the starting point of physics. One 
of the most primitive concepts is that of an object. 
The concepts of a tree, a horse, any material body, are 
creations gained on the basis of experience, though the 
impressions from which they arise are primitive in 
comparison with the world of physical phenomena. 
A cat teasing a mouse also creates, by thought, its own 
primitive reality. The fact that the cat reacts in a similar 
way toward any mouse it meets shows that it forms 
concepts and theories which are its guide through its 
own world of sense impressions. 


" Three trees" is something different from "two 
trees". Again "two trees" is different from "two 
stones". The concepts of the pure numbers 2, 3, 4, . . . , 
freed from the objects from which they arose, are 
creations of the thinking mind which describe the 
reality of our world. 

The psychological subjective feeling of time enables 
us to order our impressions, to state that one event 
precedes another. But to connect every instant of time 
with a number, by the use of a clock, to regard time as 
a one-dimensional continuum, is already an invention. 
So also are the concepts of Euclidean and non-Eucli- 
dean geometry, and our space understood as a three- 
dimensional continuum. 

Physics really began with the invention of mass, 
force, and an inertial system. These concepts are all 
free inventions. They led to the formulation of the 
mechanical point of view. For the physicist of the 
early nineteenth century, the reality of our outer 
world consisted of particles with simple forces acting 
between them and depending only on the distance. He 
tried to retain as long as possible his belief that he 
would succeed in explaining all events in nature by 
these fundamental concepts of reality. The difficulties 
connected with the deflection of the magnetic needle, 
the difficulties connected with the structure of the 
ether, induced us to create a more subtle reality. The 
important invention of the electromagnetic field ap- 
pears. A courageous scientific imagination was needed 
to realize fully that not the behaviour of bodies, but 


the behaviour of something between them, that is, the 
field, may be essential for ordering and understanding 

Later developments both destroyed old concepts 
and created new ones. Absolute time and the inertial 
co-ordinate system were abandoned by the relativity 
theory. The background for all events was no longer 
the one-dimensional time and the three-dimensional 
space continuum, but the four-dimensional time-space 
continuum, another free invention, with new transfor- 
mation properties. The inertial co-ordinate system was 
no longer needed. Every co-ordinate system is equally 
suited for the description of events in nature. 

The quantum theory again created new and essential 
features of our reality. Discontinuity replaced con- 
tinuity. Instead of laws governing individuals, prob- 
ability laws appeared. 

The reality created by modern physics is, indeed, far 
removed from the reality of the early days. But the aim 
of every physical theory still remains the same. 

With the help of physical theories we try to find our 
way through the maze of observed facts, to order and 
understand the world of our sense impressions. We 
want the observed facts to follow logically from our 
concept of reality. Without the belief that it is possible 
to grasp the reality with our theoretical constructions, 
without the belief in the inner harmony of our world, 
there could be no science. This belief is and always will 
remain the fundamental motive for all scientific crea- 
tion. Throughout all our efforts, in every dramatic 


struggle between old and new views, we recognize the 
eternal longing for understanding, the ever-firm belief 
in the harmony of our world, continually strengthened 
by the increasing obstacles to comprehension. 


Again the rich variety of facts in the realm of atomic 
phenomena forces us to invent new physical concepts. Matter 
has a granular structure; it is composed of elementary particles, 
the elementary quanta of matter. Thus, the electric charge has 
a granular structure and most important from the point of 
view of the quantum theory so has energy. Photons are the 
energy quanta of which light is composed. 

Is light a wave or a shower of photons? Is a beam of 
electrons a shower of elementary particles or a wave? These 
fundamental questions are forced upon physics by experiment. 
In seeking to answer them we have to abandon the description 
of atomic events as happenings in space and time, we have to 
retreat still further from the old mechanical view. Quantum 
physics formulates laws governing crowds and not individuals. 
Not properties but probabilities are described, not laws dis- 
closing the future of systems are formulated, but laws governing 
the changes in time of the probabilities and relating to great 
congregations of individuals. 



Absolute motion, 180, 224 
Aristotle, 6 

Black, 39, 40, 51 

Bohr, 283, 302 

Born, 302 

Brown, 63, 64 

Brownian movement, 6367 

Caloric, 43 

Change in velocity, 10, 18, 23- 
24, 28 

Classical transformation, 171 

Conductors, 74 

Constant of the motion, 49 

Continuum : 

one-dimensional, 210 
two-dimensional, 2 1 1 
three-dimensional, 212 
four-dimensional, 219 

Co-ordinate of a point, 1 68 

Co-ordinate system, 162, 163 

Copernicus, 161, 223, 224 

Corpuscles of light, 99, 100, 275 

Coulomb, 79, 86 

Crucial experiments, 44-45 

Crystal, 285 

G.S., 163 

Current, 88 
induced, 88 

de Broglie, 287, 290, 302 
Democritus, 56 
Diffraction : 

of electronic wave, 293 

Diffraction (continued) 

of light, 1 1 9 

of X-rays, 286 
Dipole : 

electric, 84 

magnetic, 85 
Dirac, 302 
Dispersion, 102, 117 
Dynamic picture of motion, 216 

Electric : 

charge, 80-82 

current, 88 

potential, 80-82 

substances, 74 
Electromagnetic : 

field, 151 

theory of light, 157 

wave, 154 

Electronic wave, 292 
Electrons, 268 
Electroscope, 71 
Elementary : 

magnetic dipoles, 85 

particles, 206 

quantum, 264 
Elements of a battery, 88 
Energy : 

kinetic, 49, 50 

level, 282 

mechanical, 51 

potential, 49, 50 
Ether, 112, 115, 120, 123-126, 

172, 175, 176, 179, 1 80- 




Faraday, 129, 142 
Field, 131 

representation, 131 

static, 141 

structure of, 149, 152 
Fizeau, 96 

Frame of reference, 1 63 
Fresnel, 118 
Force, u, 19, 24, 28 

lines, 130 

matter, 56 


Heat, 38, 41, 42 

capacity, 41 

energy, 50-55 

specific, 41 

substance, 42, 43 
Heisenberg, 302 
Helmholtz, 58, 59 
Hertz, 129, 156 
Huygens, no, in 

Induced current, 142 

Inertial mass, 36, 227, 230 
system, 166, 220, 221 
system, local, 228, 229 

Insulators, 71, 74 

Invariant, 170 

Joule, 51, 52, 53 
Kinetic theory, 59-67 

Law of gravitation, 30 

of inertia, 8, 160 

of motion, 31 
Leibnitz, 25 

Light, bending in gravitational 
field, 234, 252 

homogeneous, 103 

quanta, 275 

substance, 102-104 

white, 101 
Lorentz transformation, 198- 


Galilean relativity principle, 165 

Galileo, 5, 7, 8, 9, 39, 56, 94, Mass, 34 

Galvani, 88 
Generalization, 20 
General relativity, 36, 224 
Gravitational mass, 36, 227, 230 

energy, 208 

of one electron, 270 

of one hydrogen atom, 266 

of one hydrogen molecule, 

67, 265 

Matter = energy, 54 
field, 256 

Maxwell, 129 

Maxwell's equations, 148, 149, 

150, 153 
Mayer, 51 
Mechanical equivalent of heat, 

Mechanical view, 59, 87, 92, 

120, 124, 157 
Mercury, 253 
Metric properties, 246 
Michelson, 97, 183 
Molecules, 59 

number of, 66 
Morley, 183 

Newton, 5, 8, 9, 25, 79, 92, 100, 


Nodes, 288 
Nuclear physics, 272 

Nucleus, 271 
Oersted, go, 91 

Photoelectric effect, 273 
Photons, 275 

ultraviolet, 284 
Planck, 275 
Pole, magnetic, 83 
Principia, 1 1 
Probability, 298 

wave, 304 
Ptolemy, 223, 224 

Radioactive disintegration, 299 

matter, 206 
Radium, 206 
Rate of exchange, 52 
Rectilinear motion, 12 

propagation of light, 97, 99, 


Reflection of light, 99 
Refraction, 98, 99, 115, 116 
Relative uniform motion, 180 
Relativity, 186 

general, 36, 224 

special, 224 
Rest mass, 205 
Roemer, 96 
Rowland, 92 
Rumford, 45, 47, 51 
Rutherford, 272 

Schrodinger, 287, 302 
Sodium, 103 
Solenoid, 136 
Special relativity, 224 
Spectral lines, 280 

INDEX 319 

Spectroscope, 281 
Spectrum, visible, 102, 284 
Static picture of motion, 216 
Statistics, 298 
Synchronized clocks, 190, 191 

Temperature, 38-40 
Test body, 130 
Thomson, J. J., 268 
Tourmaline crystal, 121 
Transformation laws, 169 
Two New Sciences , 10, 94 

Uniform motion, 9 

Vectors, 12-19 
Velocity of electromagnetic 
wave, 155 

of light, 97 

vector, 21 
Volta, 88, 89 
Voltaic battery, 88 

Wave, 104 

length, 1 06, 117 

longitudinal, 108, 121 

plane, no 

spherical, 109 

standing, 288 

theory of light, no 

transverse, 108, 121 

velocity, 106 

Weightless substances, 43, 

X-rays, 284-286 
Young, 118